(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 543288, 19832]*) (*NotebookOutlinePosition[ 544396, 19865]*) (* CellTagsIndexPosition[ 544352, 19861]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["7. Taylor-Polynome", FontFamily->"Arial", FontSize->14, FontWeight->"Bold"]], "Section"], Cell[TextData[{ "Materialien zur Vorlesung ", StyleBox["Elementare Analysis", FontWeight->"Bold"], ", Wintersemester 2003 / 4\n" }], "SmallText"], Cell[TextData[{ "Zuerst soll untersucht werden, wie die Koeffizienten einer Polynomfunktion \ mit ihren Ableitungen zusammenh\[ADoubleDot]ngen.\n\nF\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[Element] \[DoubleStruckCapitalR]\)]], " sei also mit reellen ", Cell[BoxData[ \(TraditionalForm\`a\_\[Nu]\ , \ \[Nu]\ = \ 0, 1, 2, \[Ellipsis], n\)]], " und ", Cell[BoxData[ \(TraditionalForm\`\(\(a\_n \[NotEqual] 0\)\(\ \ \)\(,\)\)\)]], "\n\t\t", Cell[BoxData[ \(TraditionalForm\`p( x) := \(\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\) \(a\_\[Nu]\) x\^\[Nu] = a\_0 + \(a\_1\) x + \(a\_2\) x\^2 + \[CenterEllipsis] + \(a\_n\) \(\(x\^n\)\(\ \ \)\(.\)\)\)\ \)]], "\nDann gilt ", Cell[BoxData[ \(TraditionalForm\`p(0) = \(\(a\_0\)\(\ \)\(.\)\)\)]], " Nun wird ", Cell[BoxData[ \(TraditionalForm\`p\)]], " einmal abgeleitet:\n\t\t", Cell[BoxData[ \(TraditionalForm\`p' \((x)\) := \(0 + a\_1 + 2 \( a\_2\) x + 3 \( a\_3\) x\^2 + \[CenterEllipsis] + n\ \(a\_n\) x\^\(n - 1\)\ \ und\ \ p' \((0)\) = \(\(a\_1\)\(\ \ \)\(.\)\)\)\ \)]], "\n", Cell[BoxData[ \(TraditionalForm\`p\)]], " wird zweimal abgeleitet:\n\t\t", Cell[BoxData[ \(TraditionalForm\`p'' \((x)\) := 0 + 0 + 2 a\_2 + 3\[CenterDot]2 \( a\_3\) x\ + 4\[CenterDot]3 x\^2 + \[CenterEllipsis] + n\[CenterDot]\ \((n - 1)\) \(a\_n\) x\^\(n - 2\)\ \ , \ p'' \((0)\) = 2 \(\( a\_2\)\(\ \)\(.\)\)\)]], "\nInduktiv folgt leicht f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`k \[LessEqual] n\)]], "\n\n\t\t", Cell[BoxData[ \(TraditionalForm\`\(p\^\((k)\)\)(x)\)]], "\t", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\(\(\(\(k\[CenterDot]\((k - 1)\)\)\[CenterDot]\[CenterDot]\[CenterDot]\((k - k + 1)\)\)\[CenterDot]a\_k\)\[CenterDot]x\^0\)\(\ \)\)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[CapitalSigma]\+\(\[Nu] = k + 1\)\%n\) \[Nu]\[CenterDot]\((\[Nu] - 1)\)\[CenterDot]\[CenterDot]\[CenterDot]\((\[Nu] - k + 1)\) \(a\_\[Nu]\) x\^\(\[Nu] - k\)\)\(\ \)\(,\)\)\)]], "\n\t\t\t", Cell[BoxData[ \(TraditionalForm\`\(\(\(=\)\(\ \)\(\(\[CapitalSigma]\+\(\[Nu] = k\)\%n\) \[Nu]\[CenterDot]\((\[Nu] - 1)\)\[CenterDot]\[CenterDot]\[CenterDot]\((\[Nu] - k + 1)\) \(a\_\[Nu]\) x\^\(\[Nu] - k\)\)\)\(\ \ \)\(,\)\)\)]], "\n\t\t\t\nworaus sofort ", Cell[BoxData[ \(TraditionalForm\`\(p\^\((k)\)\)(0) = k\[CenterDot]\((k - 1)\)\[CenterDot]\[CenterDot]\[CenterDot]1\[CenterDot]\ a\_k\)]], " bzw. ", Cell[BoxData[ FormBox[ RowBox[{\(a\_k\), "=", FractionBox[ StyleBox[\(\(p\^\((k)\)\) (0)\), FontSize->10], StyleBox[\(k!\), FontSize->10]]}], TraditionalForm]]], " folgt.\n\nDies ist der gesuchte Zusammenhang zwischen den Ableitungen \ einer Polynomfunktion und ihren Koeffizienten." }], "Text"], Cell[TextData[{ StyleBox["\nSatz 7.1", FontFamily->"Arial", FontVariations->{"Underline"->True}], "\nF\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[Element] \[DoubleStruckCapitalR]\)]], " sei mit reellen ", Cell[BoxData[ \(TraditionalForm\`a\_\[Nu]\ , \ \[Nu]\ = \ 0, 1, 2, \[Ellipsis], n\)]], " und ", Cell[BoxData[ \(TraditionalForm\`\(\(a\_n \[NotEqual] 0\)\(\ \ \)\(,\)\)\)]], "\n\n\t\t", Cell[BoxData[ \(TraditionalForm\`p( x) := \(\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\) \(a\_\[Nu]\) x\^\[Nu] = a\_0 + \(a\_1\) x + \(a\_2\) x\^2 + \[CenterEllipsis] + \(a\_n\) \(\(x\^n\)\(\ \ \)\(.\)\)\)\ \)]], "\n\t\t\nDann gilt ", Cell[BoxData[ FormBox[ RowBox[{\(a\_\[Nu]\), "=", FractionBox[ RowBox[{ StyleBox[\(p\^\((\[Nu])\)\), FontSize->10], StyleBox["(", FontSize->10], StyleBox["0", FontSize->10], StyleBox[")", FontSize->10]}], StyleBox[\(\[Nu]!\), FontSize->10]]}], TraditionalForm]]], " und damit ", Cell[BoxData[ FormBox[ RowBox[{\(p(x)\), ":=", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\), FractionBox[ RowBox[{ StyleBox[\(p\^\((\[Nu])\)\), FontSize->10], StyleBox["(", FontSize->10], StyleBox["0", FontSize->10], StyleBox[")", FontSize->10]}], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(x\^\[Nu]\)\(\ \ \)\(.\)\)}]}], TraditionalForm]]] }], "Text"], Cell["\[FilledSmallSquare]", "Text", TextAlignment->Right], Cell[TextData[{ "Von dieser Beobachtung ausgehend soll nun entsprechend analysiert werden, \ wie bei anderen Funktionen Beziehungen zwischen den Ableitungen und \ sogenannten N\[ADoubleDot]herungspolynomfunktionen hergestellt werden k\ \[ODoubleDot]nnen.\nAlso: In welcher Beziehung stehen f\[UDoubleDot]r eine \ geeignete Funktion ", StyleBox["f", FontSlant->"Italic"], " die Zahlen\n\n\t\t", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " und ", Cell[BoxData[ FormBox[ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\), FractionBox[ RowBox[{ StyleBox[\(f\^\((\[Nu])\)\), FontSize->10], "(", StyleBox["0", FontSize->10], StyleBox[")", FontSize->10]}], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(x\^\[Nu]\)\(\ \ \)\(.\)\)}], TraditionalForm]]], "\n\t\t\n", Cell[BoxData[ \(TraditionalForm\`f\)]], " muss dann selbstverst\[ADoubleDot]ndlich ", StyleBox["n", FontSlant->"Italic"], "-mal differenzierbar sein und etwa auf einem Intervall ", Cell[BoxData[ \(TraditionalForm\`I := \([a, b]\)\)]], " definiert sein. \n\nEs seien nun ", Cell[BoxData[ \(TraditionalForm\`\[Alpha], \[Beta] \[Element] I\)]], " und ", Cell[BoxData[ \(TraditionalForm\`\[Alpha] < \[Beta]\)]], " . Wir betrachten f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[Element] I\)]], " die Funktion ", Cell[BoxData[ \(TraditionalForm\`h\)]], " mit\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{\(h(x)\), ":=", " ", RowBox[{\(f(x)\), "+", RowBox[{ FractionBox[ StyleBox[\((\[Beta]\ - \ x)\), FontSize->10], StyleBox[\(1!\), FontSize->10]], \(\(f\^\[Prime]\)(x)\)}], " ", "+", RowBox[{ FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^2\), FontSize->10], StyleBox[\(2!\), FontSize->10]], \(\(f\^\[DoublePrime]\)(x)\)}], "+", "\[CenterEllipsis]", "+", " ", RowBox[{ FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\(n - 2\)\), FontSize->10], StyleBox[\(\((n\ - \ 2)\)!\), FontSize->10]], \(\(f\^\((n - 1)\)\)(x)\)}], " ", "+", " ", RowBox[{ FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\(n - 1\)\), FontSize->10], StyleBox[\(\((n\ - \ 1)\)!\), FontSize->10]], \(\(f\^\((n - 1)\)\)(x)\)}]}]}], TraditionalForm]]], "\nund leiten ", Cell[BoxData[ \(TraditionalForm\`h\)]], " flei\[SZ]ig ab:\n\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(h' \((x)\)\), "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\(n - 1\)\), " ", FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(f\^\((\[Nu])\)\)(x)\)}], ")"}], "\[Prime]"], " ", "=", RowBox[{ RowBox[{\(\(\(\[CapitalSigma]\)\(\ \ \)\)\+\(\[Nu] = 0\)\%\(n - 1\)\), " ", SuperscriptBox[ RowBox[{"(", FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], ")"}], "\[Prime]"], \(\(f\^\((\[Nu])\)\)(x)\)}], " ", "+", " ", RowBox[{\(\(\(\[CapitalSigma]\)\(\ \ \)\)\+\(\[Nu] = 0\)\%\(n - 1\)\), " ", FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(f\^\((\[Nu] + 1)\)\)(x)\)}]}]}]}], " ", "\[IndentingNewLine]"}], TraditionalForm]]], "\n\t\t\t", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ RowBox[{\(\(\(\[CapitalSigma]\)\(\ \)\)\+\(\[Nu] = 0\)\%\(n - 1\)\), FractionBox[ StyleBox[\(\(-\[Nu]\)\[CenterDot]\((\[Beta]\ - \ x)\)\^\(\ \[Nu] - 1\)\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(f\^\((\[Nu])\)\)(x)\)}], " ", "+", RowBox[{\(\(\(\[CapitalSigma]\)\(\ \ \)\)\+\(\[Nu] = 0\)\%\(n - 1\)\), " ", FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(f\^\((\[Nu] + 1)\)\)(x)\), " "}]}]}], TraditionalForm]]], "\n\t\t\t\n\t\t\t", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ RowBox[{\(\(\(\[CapitalSigma]\)\(\ \)\)\+\(\[Nu] = 1\)\%\(n - 1\)\), FractionBox[ StyleBox[\(-\((\[Beta]\ - \ x)\)\^\(\[Nu] - 1\)\), FontSize->10], StyleBox[\(\((\[Nu]\ - \ 1)\)!\), FontSize->10]], \(\(f\^\((\[Nu])\)\)(x)\)}], " ", "+", RowBox[{\(\(\(\[CapitalSigma]\)\(\ \ \)\)\+\(\[Nu] = 0\)\%\(n - 1\)\), " ", FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(f\^\((\[Nu] + 1)\)\)(x)\), " "}]}]}], TraditionalForm]]], "\n\t\t\t\n\t\t\t ", Cell[BoxData[ \(TraditionalForm\` = \)]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\(\(\[CapitalSigma]\)\(\ \)\)\+\(\[Nu] = 0\)\%\(n - 2\)\), FractionBox[ StyleBox[\(-\((\[Beta]\ - \ x)\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(f\^\((\[Nu] + 1)\)\)(x)\)}], " ", "+", RowBox[{\(\(\(\[CapitalSigma]\)\(\ \ \)\)\+\(\[Nu] = 0\)\%\(n - 1\)\), " ", FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], \(\(f\^\((\[Nu] + 1)\)\)(x)\), " "}]}], TraditionalForm]]], "\n\t\t\t \n\t\t\t ", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FractionBox[ StyleBox[\(\((\[Beta]\ - \ x)\)\^\(n - 1\)\), FontSize->10], StyleBox[\(\((n\ - \ 1)\)!\), FontSize->10]], \(\(\(f\^\((n)\)\)( x)\)\(\ \ \)\(.\)\(\ \ \)\)}]}], TraditionalForm]]], "\n\n\nJetzt werde ", Cell[BoxData[ FormBox[ RowBox[{"\[Sigma]", ":=", FractionBox[ StyleBox[\(f(\[Beta])\ - \ h(\[Alpha])\), FontSize->10], StyleBox[\(\((\[Beta]\ - \ \[Alpha])\)\^n\), FontSize->10]]}], TraditionalForm]]], " gesetzt. Dann gilt mit ", Cell[BoxData[ \(TraditionalForm\`g(x) := f(\[Beta]) - h(x) - \[Sigma]\[CenterDot]\((\[Beta] - x)\)\^n\)]], "\n\n\t", Cell[BoxData[ FormBox[ RowBox[{\(g(\[Alpha])\), " ", "=", RowBox[{ RowBox[{\(f(\[Beta])\), "-", \(h(\[Alpha])\), "-", RowBox[{ FractionBox[ StyleBox[\(f(\[Beta])\ - \ h(\[Alpha])\), FontSize->10], StyleBox[\(\((\[Beta]\ - \ \[Alpha])\)\^n\), FontSize->10]], \(\((\[Beta] - \[Alpha])\)\^n\)}]}], "=", "0"}]}], TraditionalForm]]], "\n\t\nund da ", Cell[BoxData[ \(TraditionalForm\`h(\[Beta]) = f(\[Beta])\)]], ", auch ", Cell[BoxData[ \(TraditionalForm\`g(\[Beta]) = \(f(\[Beta]) - f(\[Beta]) - \[Sigma]\[CenterDot]\((\[Beta] - \[Beta])\)\^n = \ \(\(0\)\(\ \ \)\(.\)\)\)\)]], "\n\t\nAlso ist mit ", StyleBox["f", FontSlant->"Italic"], " auch ", StyleBox["g", FontSlant->"Italic"], " differenzierbar, und da ", StyleBox["f", FontSlant->"Italic"], " ", StyleBox["n", FontSlant->"Italic"], "-mal differenzierbar ist, folgt mit dem Satz von Rolle:\nEs gibt ein ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Gamma]", "\[Element]", RowBox[{ RowBox[{ StyleBox["[", FontSize->2, FontColor->GrayLevel[1]], "]"}], "\[Alpha]"}]}], ",", RowBox[{"\[Beta]", " ", "[", StyleBox["]", FontColor->GrayLevel[1]]}], ",", \(so\ dass\)}], TraditionalForm]]], " ", Cell[BoxData[ \(TraditionalForm\`\(\(\(g\^\[Prime]\)(\[Gamma]) = \(\(-\(\(h\^\[Prime]\ \)(\[Gamma])\)\) + n\[CenterDot]\[Sigma]\[CenterDot]\((\[Beta] - \[Gamma])\)\^\(n - \ 1\) = 0\)\)\(\ \ \)\(,\)\)\)]], "\nworaus nun \n\t", Cell[BoxData[ FormBox[ RowBox[{"\[Sigma]", "=", RowBox[{ FractionBox[ RowBox[{ StyleBox[\(h\^\[Prime]\), FontSize->10], "(", StyleBox["\[Gamma]", FontSize->10], StyleBox[")", FontSize->10]}], StyleBox[\(n\[CenterDot]\((\[Beta]\ - \ \[Gamma])\)\^\(n - 1\)\ \), FontSize->10]], "=", RowBox[{ FractionBox[ StyleBox[ RowBox[{\(\((\[Beta]\ - \ \[Gamma])\)\^\(n - 1\)\), RowBox[{ SuperscriptBox[ StyleBox["h", FontSize->10], \((n)\)], "(", "\[Gamma]", ")"}]}], FontSize->10], StyleBox[ RowBox[{\(\((n\ - \ 1)\)!\), "\[CenterDot]", "n", "\[CenterDot]", SuperscriptBox[ RowBox[{"(", RowBox[{"\[Beta]", StyleBox[" ", FontSize->10], "-", " ", "\[Gamma]"}], ")"}], \(n - 1\)]}], FontSize->10]], "=", FractionBox[ RowBox[{ StyleBox[\(f\^\(\(\ \)\((n)\)\)\), FontSize->10], "(", StyleBox["\[Gamma]", FontSize->10], StyleBox[")", FontSize->10]}], StyleBox[\(n!\), FontSize->10]]}]}]}], TraditionalForm]]], "\nfolgt.\nAlso ist ", Cell[BoxData[ FormBox[ RowBox[{\(g(x)\), "=", RowBox[{\(f(\[Beta])\), "-", \(h(x)\), "-", RowBox[{ FractionBox[ RowBox[{ StyleBox[\(f\^\(\(\ \)\((n)\)\)\), FontSize->10], "(", StyleBox["\[Gamma]", FontSize->10], StyleBox[")", FontSize->10]}], StyleBox[\(n!\), FontSize-> 10]], \(\(\((\[Beta] - x)\)\^n\)\(\ \ \)\(.\)\(\ \)\)}]}]}], TraditionalForm]]], "\n\nNun war ", Cell[BoxData[ \(TraditionalForm\`\(\(g(\[Alpha]) = 0\)\(\ \)\(,\)\)\)]], " und damit folgt\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(f(\[Beta])\), "=", RowBox[{\(h(\[Alpha])\), "+", RowBox[{ FractionBox[ RowBox[{ StyleBox[\(f\^\(\(\ \)\((n)\)\)\), FontSize->10], "(", StyleBox["\[Gamma]", FontSize->10], StyleBox[")", FontSize->10]}], StyleBox[\(n!\), FontSize->10]], \(\((\[Beta] - \[Alpha])\)\^n\)}]}]}], " ", ","}], TraditionalForm]]], "\nd.h.\n", Cell[BoxData[ FormBox[ RowBox[{\(f(\[Beta])\), "=", RowBox[{\(f(\[Alpha])\), "+", RowBox[{ FractionBox[ StyleBox[ RowBox[{"(", RowBox[{"\[Beta]", " ", "-", " ", StyleBox["\[Alpha]", FontSize->10]}], ")"}], FontSize->10], StyleBox[\(1!\), FontSize->10]], \(\(f\^\[Prime]\)(\[Alpha])\)}], " ", "+", RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"\[Beta]", " ", "-", " ", StyleBox["\[Alpha]", FontSize->10]}], ")"}], "2"], FontSize->10], StyleBox[\(2!\), FontSize->10]], \(\(f\^\[DoublePrime]\)(\[Alpha])\)}], "+", "\[CenterEllipsis]", "+", RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"\[Beta]", " ", "-", " ", StyleBox["\[Alpha]", FontSize->10]}], ")"}], "n"], FontSize->10], StyleBox[\(n!\), FontSize-> 10]], \(\(\(f\^\((n)\)\)(\[Alpha])\)\(\ \ \)\(.\)\)}]}]}], TraditionalForm]]], "\n\nUm nun zu den \[UDoubleDot]blichen Bezeichnungen zu kommen, seien ", Cell[BoxData[ \(TraditionalForm\`I := \([a, b]\)\)]], " , ", Cell[BoxData[ \(TraditionalForm\`f : I \[Rule] \[DoubleStruckCapitalR]\)]], " , ", Cell[BoxData[ \(TraditionalForm\`f\)]], " ", Cell[BoxData[ \(TraditionalForm\`\((n + 1)\)\)]], "-mal differenzierbar. Es werde also ", Cell[BoxData[ \(TraditionalForm\`n\)]], " durch ", Cell[BoxData[ \(TraditionalForm\`\(\(n + 1\)\(\ \)\(,\)\)\)]], " \[Beta] durch ", StyleBox["x", FontSlant->"Italic"], " , \[Alpha] durch ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " und \[Gamma] durch \[Xi] ersetzt.\n\nDann gilt der \ ber\[UDoubleDot]hmte\n" }], "Text"], Cell[TextData[{ StyleBox["Satz 7.1:\n", FontFamily->"Arial", FontVariations->{"Underline"->True}], StyleBox["Satz von Taylor (1715)", FontFamily->"Arial"], StyleBox["\n", FontVariations->{"Underline"->True}], "(", ButtonBox["Brook Taylor", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Taylor.\ html"], None}, ButtonStyle->"Hyperlink"], ", 1685 - 1731, englischer Mathematiker)\nEs seien ", Cell[BoxData[ \(TraditionalForm\`I := \([a, b]\)\)]], " , ", Cell[BoxData[ \(TraditionalForm\`f : I \[Rule] \[DoubleStruckCapitalR]\)]], ", ", StyleBox["f", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\((n + 1)\)\)]], "-mal differenzierbar und ", Cell[BoxData[ \(TraditionalForm\`x, x\_0 \[Element] \(\(I\)\(\ \)\(.\)\)\)]], " Dann gilt mit einem \[Xi] zwischen ", Cell[BoxData[ \(TraditionalForm\`x\ \ und\ \ \ x\_0\)]], "\n\n\t", Cell[BoxData[ FormBox[ RowBox[{\(f(x)\), "=", RowBox[{\(f(x\_0)\), "+", RowBox[{ FractionBox[ StyleBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], FontSize->10], StyleBox[\(1!\), FontSize->10]], \(f'\), \((x\_0)\)}], " ", "+", RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "2"], FontSize->10], StyleBox[\(2!\), FontSize->10]], \(f''\), \((x\_0)\)}], "+", "\[CenterEllipsis]", "+", RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "n"], FontSize->10], StyleBox[\(n!\), FontSize->10]], \(\(f\^\((n)\)\)(x\_0)\)}], " ", "+", " ", RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], \(n + 1\)], FontSize->10], StyleBox[\(\((n\ + \ 1)\)!\), FontSize-> 10]], \(\(\(f\^\((n + 1)\)\)(\[Xi])\)\(\ \ \)\(.\)\)}]}]}], TraditionalForm]]] }], "Text"], Cell["\[FilledSmallSquare]", "Text", TextAlignment->Right], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\), " ", FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], " ", ")"}], "\[Nu]"], FontSize->10], \(\[Nu]!\)], \(\(f\^\((\[Nu])\)\)(x\_0)\)}], TraditionalForm]]], " hei\[SZ]t das ", Cell[BoxData[ \(TraditionalForm\`n\)]], "-te ", StyleBox["Taylor-Polynom ", FontFamily->"Arial", FontSize->10], "von ", Cell[BoxData[ \(TraditionalForm\`\(\(f\)\(\ \)\(.\)\)\)]] }], "Text"], Cell[TextData[{ "Nun soll f\[UDoubleDot]r einige wichtige Funktionen die jeweiligen \ Taylor-Polynome bzw. die Taylor-Reihen berechnet werden. Man spricht auch von \ ", StyleBox["Taylor-Entwicklungen", FontFamily->"Arial", FontSize->10], ".\n\nFr\[UDoubleDot]her wurde schon bewiesen: F\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[NotEqual] 1\)]], " gilt\n\t\t", Cell[BoxData[ FormBox[ RowBox[{\(\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\) x\^\[Nu]\), "=", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ - \ x\), FontSize->10]], "-", FractionBox[ StyleBox[\(x\^\(n + 1\)\), FontSize->10], StyleBox[\(1\ - \ x\), FontSize->10]], " "}]}], TraditionalForm]]], " bzw. ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ - \ x\), FontSize->10]], "=", RowBox[{\(\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\) x\^\[Nu]\), "+", RowBox[{ FractionBox[ StyleBox[\(x\^\(n + 1\)\), FontSize->10], StyleBox[\(1\ - \ x\), FontSize->10]], " ", "."}]}]}], TraditionalForm]]], "\n\nGilt nun noch ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(\ \)\(x\)\(\ \)\(|\)\(\(<\)\(1\)\)\)\)]], ", so ist ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\(\(lim\)\(\ \)\)\+\(n \[Rule] \[Infinity]\)\), RowBox[{"(", FractionBox[ StyleBox[\(x\^\(n + 1\)\), FontSize->10], StyleBox[\(1\ - \ x\), FontSize->10]], ")"}]}], "=", \(\(0\)\(\ \ \)\(.\)\)}], TraditionalForm]]], "\nDaher folgt" }], "Text"], Cell[TextData[{ StyleBox["Satz 7.2:\n", FontFamily->"Arial", FontVariations->{"Underline"->True}], StyleBox["F\[UDoubleDot]r ", FontVariations->{"CompatibilityType"->0}], " ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(\ \)\(x\)\(\ \)\(|\)\(\(<\)\(1\)\)\)\)]], " gilt 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Es existiere eine Konstante ", Cell[BoxData[ \(TraditionalForm\`\(\(M > 0\)\(\ \)\(,\)\)\)]], " so dass ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\ \(f\^\((n)\)\)( t)\ \[RightBracketingBar] \[LessEqual] M\)]], " f\[UDoubleDot]r alle ", Cell[BoxData[ \(TraditionalForm\`t \[Element] I\)]], " und alle ", Cell[BoxData[ \(TraditionalForm\`n \[Element] \(\(\[DoubleStruckCapitalN]\)\(\ \ \ \)\(.\)\)\)]], " \nDann gilt f\[UDoubleDot]r jedes betrachtete ", Cell[BoxData[ \(TraditionalForm\`\(\(x\_0\)\(\[Element]\)\(I\)\(\ \)\)\)]], " und alle ", Cell[BoxData[ \(TraditionalForm\`x \[Element] I\)]], " \n\n\t", Cell[BoxData[ FormBox[ RowBox[{\(f(x)\), "=", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), \(\(f\^\((\[Nu])\)\)(x\_0)\), " ", RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "\[Nu]"], FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], " ", "."}]}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ StyleBox["Beweis:\n", FontFamily->"Arial"], "F\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x = x\_0\)]], " ist die Behauptung trivial. Es sei also nun ", Cell[BoxData[ \(TraditionalForm\`x \[NotEqual] \(\(x\_0\)\(\ \)\(.\)\)\)]], "\nNach Satz 7.1 gilt f\[UDoubleDot]r jedes ", Cell[BoxData[ \(TraditionalForm\`n \[Element] \[DoubleStruckCapitalN]\)]], " mit einem \[Xi] zwischen ", Cell[BoxData[ \(TraditionalForm\`x\ \ und\ \ x\_0\)]], " \n\n\t", Cell[BoxData[ FormBox[ RowBox[{\(f(x)\), "=", RowBox[{ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\), \(\(f\^\((\[Nu])\)\)(x\_0)\), " ", FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "\[Nu]"], FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]]}], " ", "+", RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], \(n + 1\)], FontSize->10], StyleBox[\(\((n\ + \ 1)\)!\), FontSize-> 10]], \(\(\(f\^\((n + 1)\)\)(\[Xi])\)\(\ \ \)\(.\)\)}]}]}], TraditionalForm]]], "\n\t\nNach Voraussetzung ist ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\ \ \(f\^\((n + 1)\)\)(\[Xi])\ \[RightBracketingBar] \[LessEqual] \(\(M\)\ \(\ \)\(.\)\)\)]], " Es bleibt also zu zeigen, dass ", Cell[BoxData[ FormBox[ RowBox[{"(", FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], \(n + 1\)], FontSize->10], StyleBox[\(\((n\ + \ 1)\)!\), FontSize->10]], ")"}], TraditionalForm]]], " eine Nullfolge ist. \nDiese Behauptung kann elementar bewiesen werden! \ Hier soll sie aber nach einer Manier bewiesen werden, die dem gleicht, was \ \[UDoubleDot]blicherweise bezeichnet wird als: Mit Kanonen auf Spatzen schie\ \[SZ]en. Doch dieses Verfahren ist hier bequem und zus\[ADoubleDot]tzlich \ lehrreich:\n\nDie Reihe ", Cell[BoxData[ FormBox[ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), " ", FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "\[Nu]"], FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], " "}], TraditionalForm]]], "wird mit dem Quotientenkriterium auf Konvergenz untersucht: \nEs sei also \ \t", Cell[BoxData[ FormBox[ RowBox[{\(a\_n\), ":=", FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "n"], FontSize->10], StyleBox[\(n!\), FontSize->10]]}], TraditionalForm]]], ".\n\nDann ist ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ StyleBox[\(a\_\(n + 1\)\), FontSize->10], StyleBox[\(a\_n\), FontSize->10]], "=", RowBox[{ RowBox[{ FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], \(n + 1\)], FontSize->10], StyleBox[\(\((n\ + \ 1)\)!\), FontSize->10]], "\[CenterDot]", FractionBox[ StyleBox[\(n!\), FontSize->10], StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "n"], FontSize->10]]}], "=", RowBox[{ FractionBox[ StyleBox[\(x\ - \ x\_0\), FontSize->10], StyleBox[\(n\ + \ 1\), FontSize->10]], "."}]}]}], TraditionalForm]]], " \n\nNun gibt es nach Archimedes ein ", Cell[BoxData[ \(TraditionalForm\`n\_0 \[Element] \[DoubleStruckCapitalN]\)]], " mit ", Cell[BoxData[ \(TraditionalForm\`\(\(0 < 2 \((b - a)\) < n\_0\)\(\ \)\(,\)\)\)]], " also ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", FractionBox[ StyleBox[\(b\ - \ a\), FontSize->10], StyleBox[\(n\_0\), FontSize->10]], "<", \(\(1\/2\)\(\ \)\(.\)\)}], TraditionalForm]]], "\nDamit folgt f\[UDoubleDot]r alle ", Cell[BoxData[ \(TraditionalForm\`n \[GreaterEqual] n\_0\)]], "\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"\[LeftBracketingBar]", FractionBox[ StyleBox[\(a\_\(n + 1\)\), FontSize->10], StyleBox[\(a\_n\), FontSize->10]], "\[RightBracketingBar]"}], "=", RowBox[{ RowBox[{"\[LeftBracketingBar]", FractionBox[ StyleBox[\(x\ - \ x\_0\), FontSize->10], StyleBox[\(n\ + \ 1\), FontSize->10]], "\[RightBracketingBar]"}], "<", FractionBox[ StyleBox[\(b\ - \ a\), FontSize->10], StyleBox[\(n\_\(\(0\)\(\ \)\)\), FontSize->10]], "<", \(1\/2\)}]}], " ", ","}], TraditionalForm]]], "\n\t\t\nwas jetzt die absolute Konvergenz der Reihe ", Cell[BoxData[ FormBox[ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), " ", FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "\[Nu]"], FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], " "}], TraditionalForm]]], "beweist. Nun bilden aber bekanntlich nach Satz 2.1.3 die Summanden einer \ konvergenten Reihe eine Nullfolge; also ist ", Cell[BoxData[ FormBox[ RowBox[{"(", FractionBox[ StyleBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["x", FontSize->10], " ", "-", " ", \(x\_0\)}], ")"}], "n"], FontSize->10], StyleBox[\(n!\), FontSize->10]], ")"}], TraditionalForm]]], " eine Nullfolge und der Satz ist bewiesen." }], "Text"], Cell["\[FilledSmallSquare]", "Text", TextAlignment->Right], Cell[TextData[{ StyleBox["Satz 7.4:\n", FontFamily->"Arial", FontVariations->{"Underline"->True}], StyleBox["F\[UDoubleDot]r ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`x \[Element] \[DoubleStruckCapitalR]\)]], " gilt ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(exp( x)\ = \ \[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), "TraditionalForm"], " ", RowBox[{ FractionBox[ StyleBox[\(x\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]!\), FontSize->10]], " ", "."}]}], TraditionalForm]]], "\n\n", 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Weise wird dann die Taylor-Reihe von ", Cell[BoxData[ \(TraditionalForm\`ln(1 + x)\)]], " so hergeleitet:\n\t", Cell[BoxData[ \(TraditionalForm\`ln(1 + x) = \[Integral]\+0\%x\)]], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ StyleBox["dt", FontSize->10, FontSlant->"Italic"], StyleBox[\(1\ + \ t\), FontSize->10]], " ", "=", \(\[Integral]\+0\%x\)}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), " ", \(\((\(-1\))\)\^\[Nu]\), \(t\^\[Nu]\), " ", StyleBox["dt", FontSlant->"Italic"]}], " ", "="}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\+0\%x\)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\((\(-1\))\)\^\[Nu]\), \(t\^\[Nu]\), " ", StyleBox["dt", FontSlant->"Italic"]}], " ", "=", " ", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]], \(\(x\^\(\[Nu] + 1\)\)\(\ \)\(.\)\)}]}]}], TraditionalForm]]], "\n\t\nDas Problem liegt in der Vertauschung von Integration und Summation, \ die mit den in dieser Vorlesung zur Verf\[UDoubleDot]gung stehenden Mitteln \ nicht direkt geleistet werden kann. Daher ein etwas l\[ADoubleDot]ngerer \ Beweis zu\n\n", StyleBox["Satz 7.5:\n", FontFamily->"Arial", FontVariations->{"Underline"->True}], StyleBox["F\[UDoubleDot]r ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`x\)]], " mit ", Cell[BoxData[ \(TraditionalForm\`\(-1\) < x \[LessEqual] 1\)]], " gilt\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{\(ln(1 + x)\), "=", RowBox[{ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]], " ", \(x\^\(\[Nu] + 1\)\)}], " ", "=", RowBox[{"x", "-", Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox[ StyleBox["x", FontSize->10], "2"], StyleBox["2", FontSize->10]], TraditionalForm]]]]], "+", FractionBox[ SuperscriptBox[ StyleBox["x", FontSize->10], "3"], "3"], "-", FractionBox[ SuperscriptBox[ StyleBox["x", FontSize->10], "4"], StyleBox["4", FontSize->10]], "+", \(-\(\(\[CenterEllipsis]\)\(\ \ \)\(.\)\)\)}]}]}], TraditionalForm]]], "\n\t\t\nInsbesondere folgt f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x = \(\(1\)\(\ \)\(:\)\)\)]], "\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{\(ln(2)\), "=", RowBox[{ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]]}], "=", RowBox[{"1", "-", Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->10], StyleBox["2", FontSize->10]], TraditionalForm]]]]], "+", FractionBox[ StyleBox["1", FontSize->10], "3"], "-", FractionBox[ StyleBox["1", FontSize->10], StyleBox["4", FontSize->10]], "+", \(-\(\(\[CenterEllipsis]\)\(\ \ \)\(.\)\)\)}]}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ StyleBox["Beweis:", FontFamily->"Arial"], "\nF\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`n \[Element] \[DoubleStruckCapitalN]\)]], " und ", Cell[BoxData[ \(TraditionalForm\`x > \(-1\)\)]], " sei ", Cell[BoxData[ FormBox[ RowBox[{\(\(f\_n\)(x)\), ":=", RowBox[{\(ln(1 + x)\), " ", "-", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize-> 10]], \(\(x\^\(\[Nu] + 1\)\)\(\ \)\(.\)\(\ \)\)}]}]}], TraditionalForm]]], "\nDamit gilt ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\)(0) = \(\(0\)\(\ \)\(.\)\)\)]], "\nDann ist ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\^\[Prime]\)(x)\)]], "\t", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]], "-", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]], \(\((\[Nu] + 1)\)\[CenterDot] x\^\[Nu]\)}]}], "=", RowBox[{ RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]], "-", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%n\), " ", StyleBox[ SuperscriptBox[ RowBox[{"(", StyleBox[\(-1\), FontSize->10], StyleBox[")", FontSize->10]}], "\[Nu]"], FontSize->10], \(x\^\[Nu]\)}]}], "="}]}]}], TraditionalForm]]], "\n\n\t\t\t", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]], "-", RowBox[{"(", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]], "-", FractionBox[ StyleBox[\(\((\(-x\))\)\^\(n + 1\)\), FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]]}], " ", ")"}]}], "=", FractionBox[ StyleBox[\(\((\(-x\))\)\^\(n + 1\)\), FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]]}]}], TraditionalForm]]], "\nund ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\^\[Prime]\)(0) = \(\(0\)\(\ \ \)\(.\)\)\)]], "\nNun sei ", Cell[BoxData[ \(TraditionalForm\`n\)]], " ungerade, etwa ", Cell[BoxData[ \(TraditionalForm\`n = 2 m + \(\(1\)\(\ \)\(.\)\)\)]], " \n\nDann ist ", Cell[BoxData[ FormBox[ RowBox[{\(\(f\_n\^\[Prime]\)(x)\), "=", RowBox[{ FractionBox[ StyleBox[\(\((\(-x\))\)\^\(2 m + 2\)\), FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]], "\[GreaterEqual]", "0"}]}], TraditionalForm]]], " f\[UDoubleDot]r ", Cell[BoxData[ FormBox[ RowBox[{\(x \[GreaterEqual] 0\), " ", ",", " ", RowBox[{"also", " ", "mit", " ", Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(f\_n\)(0)\)]]]], " "}]}], TraditionalForm]]], "und dem sog. ", StyleBox["horse track principle", FontSlant->"Italic"], " (siehe \[CapitalUDoubleDot]bungsaufgabe) folgt ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\)(x) \[GreaterEqual] 0\)]], " f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`\(\(x \[GreaterEqual] 0\)\(\ \)\(,\)\)\)]], " \n\nd.h.\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\(2 m + 1\)\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]], \(x\^\(\[Nu] + 1\)\)}], "\[LessEqual]", \(\(ln(1 + x)\)\(\ \ \)\(.\)\)}], TraditionalForm]]], "\n\nIst ", Cell[BoxData[ \(TraditionalForm\`n\)]], " dagegen gerade, etwa ", Cell[BoxData[ \(TraditionalForm\`\(\(n = 2 m\)\(\ \)\(,\)\)\)]], " so ist ", Cell[BoxData[ FormBox[ RowBox[{\(\(f\_n\^\[Prime]\)(x)\), "=", RowBox[{ FractionBox[ StyleBox[\(\((\(-x\))\)\^\(2 m + 1\)\), FontSize->10], StyleBox[\(1\ + \ x\), FontSize->10]], "\[LessEqual]", " ", "0"}]}], TraditionalForm]]], " f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[GreaterEqual] 0\)]], " und wieder folgt\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\(2 m\)\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]], \(x\^\(\[Nu] + 1\)\)}], "\[GreaterEqual]", " ", \(ln(1 + x)\)}], " ", ","}], TraditionalForm]]], "\nalso insgesamt\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\(2 m + 1\)\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]], \(x\^\(\[Nu] + 1\)\)}], "\[LessEqual]", \(ln(1 + x)\), " ", "\[LessEqual]", RowBox[{\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\(2 m\)\), " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(\[Nu]\ + \ 1\), FontSize->10]], \(\(x\^\(\[Nu] + 1\)\)\(\ \ \)\(.\)\)}]}], TraditionalForm]]], "\n\t\t\nDie Differenz zwischen der linken und rechten Summe betr\ \[ADoubleDot]gt 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.11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29517 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29029 L .14798 .28461 L .15799 .27628 L .17773 .25407 L .19675 .22739 L .21504 .1995 L .23451 .17079 L .25326 .14789 L .25817 .14318 L .2635 .13888 L .26853 .13569 L .27096 .13449 L .27317 .1336 L .27542 .13291 L .27665 .13262 L .27722 .13251 L .27783 .13241 L Mistroke .2784 .13233 L .27891 .13228 L .27942 .13223 L .27991 .1322 L .28048 .13218 L .281 .13218 L .28158 .13219 L .28218 .13221 L .2828 .13226 L .28336 .13233 L .28401 .13242 L .28463 .13253 L .28572 .13277 L .28691 .13311 L .2896 .13418 L .29209 .13557 L .29672 .13921 L .30107 .14401 L .30583 .15096 L .31096 .16063 L .3206 .18592 L .32544 .20266 L .33073 .22452 L .34017 .2742 L .35023 .34545 L Mfstroke .35023 .34545 m .356 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22715 L .21504 .19876 L .23451 .16866 L .25326 .14245 L .26281 .13087 L .27317 .12001 L .28231 .11203 L .29237 .10502 L .30191 .09999 L .31201 .09613 L .32213 .09327 L .33137 .09087 L .33594 .08951 L Mistroke .34093 .08769 L .34564 .08548 L .35001 .08284 L .35509 .07881 L .35988 .07381 L .36417 .06808 L .36881 .06028 L .37339 .05061 L .3784 .03732 L .3837 .01962 L Mfstroke .3837 .01962 m .38843 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22715 L .21504 .19876 L .23451 .16866 L .25326 .14245 L .26281 .13087 L .27317 .12001 L .28231 .11203 L .29237 .10502 L .30191 .09999 L .31201 .09613 L .32213 .09327 L .33137 .09087 L .33594 .08951 L Mistroke .34093 .08769 L .34564 .08548 L .35001 .08284 L .35509 .07881 L .35988 .07381 L .36417 .06808 L .36881 .06028 L .37339 .05061 L .3784 .03732 L .3837 .01962 L Mfstroke .3837 .01962 m .38843 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16882 L .25326 .14295 L .26281 .13171 L .27317 .12148 L .27802 .11748 L .28316 .11382 L .28797 .11098 L .29237 .10889 L .2969 .10725 L .29939 .10658 L .30054 .10633 L Mistroke .30176 .1061 L .30291 .10592 L .30394 .10579 L .30499 .10569 L .30545 .10566 L .30597 .10563 L .30653 .1056 L .30712 .10558 L .30766 .10557 L .30817 .10557 L .30877 .10558 L .30932 .1056 L .30995 .10563 L .31055 .10567 L .31169 .10577 L .31276 .1059 L .3152 .10631 L .31757 .10687 L .32013 .10766 L .32475 .10957 L .32898 .11186 L .33808 .11859 L .34802 .12882 L .35745 .1414 L .36751 .15814 L .37709 .17759 L .38746 .20317 L .40667 .26707 L .41623 .31003 L .42634 .3671 L Mfstroke .42634 .3671 m .43092 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16882 L .25326 .14295 L .26281 .13171 L .27317 .12148 L .27802 .11748 L .28316 .11382 L .28797 .11098 L .29237 .10889 L .2969 .10725 L .29939 .10658 L .30054 .10633 L Mistroke .30176 .1061 L .30291 .10592 L .30394 .10579 L .30499 .10569 L .30545 .10566 L .30597 .10563 L .30653 .1056 L .30712 .10558 L .30766 .10557 L .30817 .10557 L .30877 .10558 L .30932 .1056 L .30995 .10563 L .31055 .10567 L .31169 .10577 L .31276 .1059 L .3152 .10631 L .31757 .10687 L .32013 .10766 L .32475 .10957 L .32898 .11186 L .33808 .11859 L .34802 .12882 L .35745 .1414 L .36751 .15814 L .37709 .17759 L .38746 .20317 L .40667 .26707 L .41623 .31003 L .42634 .3671 L Mfstroke .42634 .3671 m .43092 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12136 L .28316 .11361 L .28792 .11073 L .29237 .10853 L .29671 .10685 L .29894 .10617 L .30137 .10558 L .30275 .10531 L Mistroke .30403 .1051 L .30525 .10494 L .30587 .10488 L .30655 .10482 L .30722 .10477 L .30783 .10474 L .30841 .10471 L .30903 .1047 L .30959 .10469 L .3101 .10469 L .31067 .1047 L .31128 .10472 L .31187 .10475 L .31251 .10479 L .31382 .10491 L .31504 .10506 L .31618 .10523 L .31874 .10574 L .32149 .10647 L .32654 .1083 L .33116 .11052 L .34026 .11635 L .3502 .12475 L .3697 .14619 L .38846 .17044 L .39804 .18288 L .40841 .1953 L .41327 .20046 L .41841 .20523 L .42323 .20891 L .42762 .21146 L .42996 .21245 L .43103 .21281 L .43217 .21312 L .43274 .21326 L .43338 .21338 L .43396 .21347 L .4345 .21354 L .43513 .21359 L .43546 .21361 L .43581 .21363 L .4364 .21363 L .43673 .21362 L .43704 .21361 L .43759 .21357 L .43818 .2135 L .43875 .21342 L .43927 .21332 L .44025 .21308 L .44129 .21275 L Mistroke .44345 .21178 L .44468 .21107 L .44584 .21028 L .44822 .20826 L .45082 .20543 L .45556 .19842 L .46017 .18898 L .46446 .17757 L .46953 .16022 L .47431 .13952 L .48325 .08715 L .48794 .05109 L .49289 .00536 L Mfstroke .49289 .00536 m .49334 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12136 L .28316 .11361 L .28792 .11073 L .29237 .10853 L .29671 .10685 L .29894 .10617 L .30137 .10558 L .30275 .10531 L Mistroke .30403 .1051 L .30525 .10494 L .30587 .10488 L .30655 .10482 L .30722 .10477 L .30783 .10474 L .30841 .10471 L .30903 .1047 L .30959 .10469 L .3101 .10469 L .31067 .1047 L .31128 .10472 L .31187 .10475 L .31251 .10479 L .31382 .10491 L .31504 .10506 L .31618 .10523 L .31874 .10574 L .32149 .10647 L .32654 .1083 L .33116 .11052 L .34026 .11635 L .3502 .12475 L .3697 .14619 L .38846 .17044 L .39804 .18288 L .40841 .1953 L .41327 .20046 L .41841 .20523 L .42323 .20891 L .42762 .21146 L .42996 .21245 L .43103 .21281 L .43217 .21312 L .43274 .21326 L .43338 .21338 L .43396 .21347 L .4345 .21354 L .43513 .21359 L .43546 .21361 L .43581 .21363 L .4364 .21363 L .43673 .21362 L .43704 .21361 L .43759 .21357 L .43818 .2135 L .43875 .21342 L .43927 .21332 L .44025 .21308 L .44129 .21275 L Mistroke .44345 .21178 L .44468 .21107 L .44584 .21028 L .44822 .20826 L .45082 .20543 L .45556 .19842 L .46017 .18898 L .46446 .17757 L .46953 .16022 L .47431 .13952 L .48325 .08715 L .48794 .05109 L .49289 .00536 L Mfstroke .49289 .00536 m .49334 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11363 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10616 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10489 L .30704 .10485 L .30768 .10481 L .30826 .10479 L .30879 .10477 L .30937 .10477 L .31 .10477 L .31032 .10477 L .31066 .10478 L .31128 .10481 L .31187 .10484 L .31251 .10488 L .31382 .10501 L .31504 .10516 L .31618 .10534 L .31874 .10587 L .32149 .10662 L .32654 .1085 L .33116 .11078 L .34026 .11677 L .3502 .12546 L .3697 .14811 L .38846 .17517 L .40841 .20697 L .42762 .2382 L .44611 .2673 L .46578 .29754 L .48472 .32974 L .49424 .34984 L .50293 .37266 L Mfstroke .50293 .37266 m .51092 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11363 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10616 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10489 L .30704 .10485 L .30768 .10481 L .30826 .10479 L .30879 .10477 L .30937 .10477 L .31 .10477 L .31032 .10477 L .31066 .10478 L .31128 .10481 L .31187 .10484 L .31251 .10488 L .31382 .10501 L .31504 .10516 L .31618 .10534 L .31874 .10587 L .32149 .10662 L .32654 .1085 L .33116 .11078 L .34026 .11677 L .3502 .12546 L .3697 .14811 L .38846 .17517 L .40841 .20697 L .42762 .2382 L .44611 .2673 L .46578 .29754 L .48472 .32974 L .49424 .34984 L .50293 .37266 L Mfstroke .50293 .37266 m .51092 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .1254 L .3697 .14793 L .38846 .17467 L .40841 .2056 L .42762 .23473 L .44611 .25918 L .45548 .26935 L .46045 .27397 L .46578 .27826 L .4709 .28168 L .47557 .28413 L .47782 .28508 L .48027 .28593 L .48258 .28655 L .48362 .28676 L .48472 .28695 L .48527 .28703 L .48587 .28711 L .48643 .28717 L .48694 .28721 L .48754 .28725 L .48787 .28726 L .48818 .28727 L .48873 .28728 L .48933 .28728 L .48985 .28727 L .49043 .28724 L .49102 .2872 L Mistroke .49159 .28715 L .49259 .28703 L .49367 .28685 L .49493 .28659 L .49631 .28622 L .49883 .28535 L .50128 .28423 L .50355 .28295 L .50863 .27919 L .51341 .27443 L .52234 .26193 L .52703 .25323 L .53199 .24223 L .54086 .21706 L .55029 .18086 L .55909 .1358 L .56887 .06892 L Mfstroke .56887 .06892 m .57641 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .1254 L .3697 .14793 L .38846 .17467 L .40841 .2056 L .42762 .23473 L .44611 .25918 L .45548 .26935 L .46045 .27397 L .46578 .27826 L .4709 .28168 L .47557 .28413 L .47782 .28508 L .48027 .28593 L .48258 .28655 L .48362 .28676 L .48472 .28695 L .48527 .28703 L .48587 .28711 L .48643 .28717 L .48694 .28721 L .48754 .28725 L .48787 .28726 L .48818 .28727 L .48873 .28728 L .48933 .28728 L .48985 .28727 L .49043 .28724 L .49102 .2872 L Mistroke .49159 .28715 L .49259 .28703 L .49367 .28685 L .49493 .28659 L .49631 .28622 L .49883 .28535 L .50128 .28423 L .50355 .28295 L .50863 .27919 L .51341 .27443 L .52234 .26193 L .52703 .25323 L .53199 .24223 L .54086 .21706 L .55029 .18086 L .55909 .1358 L .56887 .06892 L Mfstroke .56887 .06892 m .57641 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20573 L .42762 .2351 L .44611 .26012 L .45548 .27084 L .46578 .2807 L .47557 .288 L .47995 .29056 L .48472 .29284 L .48732 .29386 L .48973 .29466 L .49198 .29528 L .49447 .29583 L .49554 .29602 L .4967 .2962 L .4977 .29632 L .49879 .29644 L .49996 .29652 L .50059 .29656 L .50118 .29658 L .50174 .2966 L .50224 .29661 L .50283 .29661 L .50338 .29661 L .50393 .29659 L .50452 .29657 L Mistroke .50508 .29655 L .5056 .29652 L .50685 .29642 L .50804 .29629 L .5104 .29595 L .51297 .29545 L .51757 .29421 L .52183 .29271 L .53168 .28813 L .54062 .28298 L .55064 .2768 L .55504 .27421 L .55986 .27162 L .56421 .26965 L .5666 .26878 L .56886 .26811 L .5701 .26782 L .57144 .26757 L .57203 .26748 L .57266 .26741 L .57319 .26736 L .57378 .26732 L .57438 .2673 L .57492 .26729 L .57552 .2673 L .57615 .26733 L .5767 .26737 L .57721 .26743 L .57777 .26751 L .57837 .26761 L .57962 .26789 L .58093 .2683 L .58329 .26933 L .58461 .27008 L .58583 .2709 L .58863 .27325 L .59107 .27591 L .59329 .27888 L .5983 .28772 L .60305 .29931 L .60744 .31341 L .61226 .33329 L .6174 .36047 L .62235 .39374 L Mfstroke .62235 .39374 m .62309 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20573 L .42762 .2351 L .44611 .26012 L .45548 .27084 L .46578 .2807 L .47557 .288 L .47995 .29056 L .48472 .29284 L .48732 .29386 L .48973 .29466 L .49198 .29528 L .49447 .29583 L .49554 .29602 L .4967 .2962 L .4977 .29632 L .49879 .29644 L .49996 .29652 L .50059 .29656 L .50118 .29658 L .50174 .2966 L .50224 .29661 L .50283 .29661 L .50338 .29661 L .50393 .29659 L .50452 .29657 L Mistroke .50508 .29655 L .5056 .29652 L .50685 .29642 L .50804 .29629 L .5104 .29595 L .51297 .29545 L .51757 .29421 L .52183 .29271 L .53168 .28813 L .54062 .28298 L .55064 .2768 L .55504 .27421 L .55986 .27162 L .56421 .26965 L .5666 .26878 L .56886 .26811 L .5701 .26782 L .57144 .26757 L .57203 .26748 L .57266 .26741 L .57319 .26736 L .57378 .26732 L .57438 .2673 L .57492 .26729 L .57552 .2673 L .57615 .26733 L .5767 .26737 L .57721 .26743 L .57777 .26751 L .57837 .26761 L .57962 .26789 L .58093 .2683 L .58329 .26933 L .58461 .27008 L .58583 .2709 L .58863 .27325 L .59107 .27591 L .59329 .27888 L .5983 .28772 L .60305 .29931 L .60744 .31341 L .61226 .33329 L .6174 .36047 L .62235 .39374 L Mfstroke .62235 .39374 m .62309 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23506 L .44611 .26003 L .45548 .27069 L .46578 .28045 L .4705 .28416 L .47557 .28757 L .48032 .29022 L .48472 .29217 L .48959 .29376 L .49217 .29436 L .49347 .29459 L .4949 .2948 L .49615 .29494 L .4973 .29503 L .4979 .29506 L .49824 .29508 L .49855 .29509 L .49888 .2951 L .49924 .2951 L .49957 .2951 L .49988 .2951 L .50044 .2951 L .50105 .29508 L .50156 .29506 L Mistroke .50212 .29503 L .50273 .29499 L .50338 .29493 L .50456 .2948 L .50577 .29463 L .50707 .2944 L .50941 .29387 L .51219 .29307 L .51471 .29216 L .51968 .2899 L .52435 .28719 L .534 .27991 L .54287 .27127 L .56183 .24708 L .58007 .21723 L .59948 .17849 L .61817 .13207 L .62768 .10289 L .63803 .06432 L .64797 .01728 L Mfstroke .64797 .01728 m .65079 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23506 L .44611 .26003 L .45548 .27069 L .46578 .28045 L .4705 .28416 L .47557 .28757 L .48032 .29022 L .48472 .29217 L .48959 .29376 L .49217 .29436 L .49347 .29459 L .4949 .2948 L .49615 .29494 L .4973 .29503 L .4979 .29506 L .49824 .29508 L .49855 .29509 L .49888 .2951 L .49924 .2951 L .49957 .2951 L .49988 .2951 L .50044 .2951 L .50105 .29508 L .50156 .29506 L Mistroke .50212 .29503 L .50273 .29499 L .50338 .29493 L .50456 .2948 L .50577 .29463 L .50707 .2944 L .50941 .29387 L .51219 .29307 L .51471 .29216 L .51968 .2899 L .52435 .28719 L .534 .27991 L .54287 .27127 L .56183 .24708 L .58007 .21723 L .59948 .17849 L .61817 .13207 L .62768 .10289 L .63803 .06432 L .64797 .01728 L Mfstroke .64797 .01728 m .65079 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29027 L .48472 .29223 L .48959 .29385 L .49217 .29445 L .49348 .2947 L .4949 .29491 L .49613 .29505 L .49667 .2951 L .49727 .29515 L .49779 .29519 L .49835 .29521 L .49894 .29523 L .49949 .29525 L .49983 .29525 L .50014 .29525 L .50049 .29525 L .50085 .29524 L .50147 .29522 L Mistroke .50182 .29521 L .50214 .29519 L .5033 .29511 L .50396 .29505 L .50456 .29498 L .50685 .29465 L .50929 .29414 L .51368 .29285 L .51889 .29068 L .52363 .28814 L .53287 .28165 L .54287 .27253 L .56256 .24921 L .58152 .22219 L .59976 .19487 L .61917 .16815 L .62812 .15796 L .63284 .15341 L .63786 .14935 L .64038 .14765 L .64316 .14607 L .64569 .14492 L .64806 .14411 L .64924 .14381 L .64989 .14367 L .65049 .14357 L .65104 .14349 L .65155 .14343 L .65211 .14338 L .65271 .14335 L .65302 .14334 L .65335 .14334 L .65395 .14335 L .65428 .14337 L .65458 .14339 L .65492 .14342 L .65529 .14346 L .65594 .14355 L .65656 .14365 L .65772 .14393 L .65897 .14432 L .66014 .14479 L .66234 .14592 L .6647 .14756 L .66725 .14983 L .67186 .1554 L .67609 .16239 L .68078 .17255 L .68515 .18461 L .69025 .20238 L Mistroke .69507 .22341 L .69977 .2485 L .70487 .28178 L .71379 .3585 L Mfstroke .71379 .3585 m .7172 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29027 L .48472 .29223 L .48959 .29385 L .49217 .29445 L .49348 .2947 L .4949 .29491 L .49613 .29505 L .49667 .2951 L .49727 .29515 L .49779 .29519 L .49835 .29521 L .49894 .29523 L .49949 .29525 L .49983 .29525 L .50014 .29525 L .50049 .29525 L .50085 .29524 L .50147 .29522 L Mistroke .50182 .29521 L .50214 .29519 L .5033 .29511 L .50396 .29505 L .50456 .29498 L .50685 .29465 L .50929 .29414 L .51368 .29285 L .51889 .29068 L .52363 .28814 L .53287 .28165 L .54287 .27253 L .56256 .24921 L .58152 .22219 L .59976 .19487 L .61917 .16815 L .62812 .15796 L .63284 .15341 L .63786 .14935 L .64038 .14765 L .64316 .14607 L .64569 .14492 L .64806 .14411 L .64924 .14381 L .64989 .14367 L .65049 .14357 L .65104 .14349 L .65155 .14343 L .65211 .14338 L .65271 .14335 L .65302 .14334 L .65335 .14334 L .65395 .14335 L .65428 .14337 L .65458 .14339 L .65492 .14342 L .65529 .14346 L .65594 .14355 L .65656 .14365 L .65772 .14393 L .65897 .14432 L .66014 .14479 L .66234 .14592 L .6647 .14756 L .66725 .14983 L .67186 .1554 L .67609 .16239 L .68078 .17255 L .68515 .18461 L .69025 .20238 L Mistroke .69507 .22341 L .69977 .2485 L .70487 .28178 L .71379 .3585 L Mfstroke .71379 .3585 m .7172 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29026 L .48472 .29223 L .48959 .29384 L .49217 .29444 L .49348 .29469 L .4949 .2949 L .49613 .29504 L .49667 .29509 L .49727 .29514 L .49779 .29517 L .49835 .2952 L .49894 .29522 L .49949 .29523 L .49983 .29524 L .50014 .29524 L .50049 .29523 L .50085 .29523 L .50147 .29521 L Mistroke .50182 .29519 L .50214 .29518 L .5033 .2951 L .50396 .29503 L .50456 .29497 L .50685 .29463 L .50929 .29412 L .51368 .29282 L .51889 .29065 L .52363 .2881 L .53287 .28158 L .54287 .2724 L .56256 .24885 L .58152 .22128 L .59976 .19269 L .61917 .1628 L .63786 .13703 L .65772 .11483 L .67685 .09892 L .69526 .08673 L .70457 .08009 L .70952 .07588 L .71485 .07046 L .71992 .06407 L .72458 .05681 L .72931 .04766 L .7337 .03713 L .74279 .00715 L Mfstroke .74279 .00715 m .74438 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29026 L .48472 .29223 L .48959 .29384 L .49217 .29444 L .49348 .29469 L .4949 .2949 L .49613 .29504 L .49667 .29509 L .49727 .29514 L .49779 .29517 L .49835 .2952 L .49894 .29522 L .49949 .29523 L .49983 .29524 L .50014 .29524 L .50049 .29523 L .50085 .29523 L .50147 .29521 L Mistroke .50182 .29519 L .50214 .29518 L .5033 .2951 L .50396 .29503 L .50456 .29497 L .50685 .29463 L .50929 .29412 L .51368 .29282 L .51889 .29065 L .52363 .2881 L .53287 .28158 L .54287 .2724 L .56256 .24885 L .58152 .22128 L .59976 .19269 L .61917 .1628 L .63786 .13703 L .65772 .11483 L .67685 .09892 L .69526 .08673 L .70457 .08009 L .70952 .07588 L .71485 .07046 L .71992 .06407 L .72458 .05681 L .72931 .04766 L .7337 .03713 L .74279 .00715 L Mfstroke .74279 .00715 m .74438 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29026 L .48472 .29223 L .48959 .29384 L .49217 .29444 L .49348 .29469 L .4949 .2949 L .49613 .29504 L .49667 .29509 L .49727 .29514 L .49779 .29517 L .49835 .2952 L .49894 .29522 L .49949 .29523 L .49983 .29524 L .50014 .29524 L .50049 .29524 L .50085 .29523 L .50147 .29521 L Mistroke .50182 .2952 L .50214 .29518 L .5033 .2951 L .50396 .29504 L .50456 .29497 L .50685 .29463 L .50929 .29412 L .51368 .29283 L .51889 .29065 L .52363 .2881 L .53287 .28158 L .54287 .27241 L .56256 .24889 L .58152 .22137 L .59976 .19293 L .61917 .16344 L .63786 .13859 L .64737 .12808 L .65772 .11874 L .66254 .11522 L .66766 .1121 L .67203 .10996 L .67685 .1082 L .67922 .10756 L .68051 .10728 L .6817 .10707 L .68282 .1069 L .68383 .10678 L .68495 .10668 L .68557 .10664 L .68615 .10661 L .68673 .1066 L .68706 .10659 L .68736 .10659 L .68798 .1066 L .68833 .1066 L .68865 .10661 L .68922 .10664 L .68983 .10668 L .69095 .10678 L .69226 .10694 L .69366 .10717 L .69616 .10774 L .69874 .10852 L .70112 .10943 L .70647 .11214 L .71118 .11531 L .71616 .11948 L .72561 .12987 L .73442 .14273 L Mistroke .74421 .16116 L .75313 .18245 L .76231 .21001 L .77229 .24836 L .78181 .29602 L .7919 .36308 L Mfstroke .7919 .36308 m .79603 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29026 L .48472 .29223 L .48959 .29384 L .49217 .29444 L .49348 .29469 L .4949 .2949 L .49613 .29504 L .49667 .29509 L .49727 .29514 L .49779 .29517 L .49835 .2952 L .49894 .29522 L .49949 .29523 L .49983 .29524 L .50014 .29524 L .50049 .29524 L .50085 .29523 L .50147 .29521 L Mistroke .50182 .2952 L .50214 .29518 L .5033 .2951 L .50396 .29504 L .50456 .29497 L .50685 .29463 L .50929 .29412 L .51368 .29283 L .51889 .29065 L .52363 .2881 L .53287 .28158 L .54287 .27241 L .56256 .24889 L .58152 .22137 L .59976 .19293 L .61917 .16344 L .63786 .13859 L .64737 .12808 L .65772 .11874 L .66254 .11522 L .66766 .1121 L .67203 .10996 L .67685 .1082 L .67922 .10756 L .68051 .10728 L .6817 .10707 L .68282 .1069 L .68383 .10678 L .68495 .10668 L .68557 .10664 L .68615 .10661 L .68673 .1066 L .68706 .10659 L .68736 .10659 L .68798 .1066 L .68833 .1066 L .68865 .10661 L .68922 .10664 L .68983 .10668 L .69095 .10678 L .69226 .10694 L .69366 .10717 L .69616 .10774 L .69874 .10852 L .70112 .10943 L .70647 .11214 L .71118 .11531 L .71616 .11948 L .72561 .12987 L .73442 .14273 L Mistroke .74421 .16116 L .75313 .18245 L .76231 .21001 L .77229 .24836 L .78181 .29602 L .7919 .36308 L Mfstroke .7919 .36308 m .79603 .4 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29026 L .48472 .29223 L .48959 .29384 L .49217 .29444 L .49348 .29469 L .4949 .2949 L .49613 .29504 L .49667 .29509 L .49727 .29514 L .49779 .29517 L .49835 .2952 L .49894 .29522 L .49949 .29523 L .49983 .29524 L .50014 .29524 L .50049 .29524 L .50085 .29523 L .50147 .29521 L Mistroke .50182 .2952 L .50214 .29518 L .5033 .2951 L .50396 .29504 L .50456 .29497 L .50685 .29463 L .50929 .29412 L .51368 .29283 L .51889 .29065 L .52363 .2881 L .53287 .28158 L .54287 .27241 L .56256 .24888 L .58152 .22136 L .59976 .19291 L .61917 .16337 L .63786 .13842 L .64737 .12781 L .65772 .11828 L .66254 .11464 L .66766 .11135 L .67246 .10884 L .67685 .10704 L .68136 .10569 L .68372 .10519 L .68492 .10499 L .68621 .10481 L .68735 .10469 L .68838 .10461 L .68888 .10458 L .68942 .10456 L .68993 .10454 L .6904 .10453 L .69095 .10453 L .69146 .10454 L .69202 .10455 L .69263 .10457 L .69326 .1046 L .69386 .10464 L .69498 .10474 L .69623 .10489 L .69741 .10506 L .6996 .10548 L .70197 .10607 L .70452 .10685 L .70912 .10867 L .71335 .11077 L .72241 .11661 L .73234 .1249 L .75178 .14535 L Mistroke .77049 .16696 L .78002 .17686 L .78505 .1813 L .79038 .18513 L .79266 .18643 L .79512 .18756 L .79652 .18807 L .79781 .18844 L .79843 .18859 L .79911 .18872 L .79975 .18882 L .80034 .18889 L .80092 .18893 L .80154 .18895 L .80213 .18895 L .80267 .18893 L .80316 .18889 L .8037 .18883 L .80426 .18874 L .80479 .18864 L .80599 .18832 L .80708 .18792 L .80834 .18735 L .80954 .18666 L .81174 .18506 L .81417 .1827 L .81671 .17954 L .81909 .17585 L .82334 .1673 L .82793 .15485 L .83252 .13844 L .83754 .1151 L .84279 .08355 L .84767 .04628 L Mfstroke .84767 .04628 m .85252 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29026 L .48472 .29223 L .48959 .29384 L .49217 .29444 L .49348 .29469 L .4949 .2949 L .49613 .29504 L .49667 .29509 L .49727 .29514 L .49779 .29517 L .49835 .2952 L .49894 .29522 L .49949 .29523 L .49983 .29524 L .50014 .29524 L .50049 .29524 L .50085 .29523 L .50147 .29521 L Mistroke .50182 .2952 L .50214 .29518 L .5033 .2951 L .50396 .29504 L .50456 .29497 L .50685 .29463 L .50929 .29412 L .51368 .29283 L .51889 .29065 L .52363 .2881 L .53287 .28158 L .54287 .27241 L .56256 .24888 L .58152 .22136 L .59976 .19291 L .61917 .16337 L .63786 .13842 L .64737 .12781 L .65772 .11828 L .66254 .11464 L .66766 .11135 L .67246 .10884 L .67685 .10704 L .68136 .10569 L .68372 .10519 L .68492 .10499 L .68621 .10481 L .68735 .10469 L .68838 .10461 L .68888 .10458 L .68942 .10456 L .68993 .10454 L .6904 .10453 L .69095 .10453 L .69146 .10454 L .69202 .10455 L .69263 .10457 L .69326 .1046 L .69386 .10464 L .69498 .10474 L .69623 .10489 L .69741 .10506 L .6996 .10548 L .70197 .10607 L .70452 .10685 L .70912 .10867 L .71335 .11077 L .72241 .11661 L .73234 .1249 L .75178 .14535 L Mistroke .77049 .16696 L .78002 .17686 L .78505 .1813 L .79038 .18513 L .79266 .18643 L .79512 .18756 L .79652 .18807 L .79781 .18844 L .79843 .18859 L .79911 .18872 L .79975 .18882 L .80034 .18889 L .80092 .18893 L .80154 .18895 L .80213 .18895 L .80267 .18893 L .80316 .18889 L .8037 .18883 L .80426 .18874 L .80479 .18864 L .80599 .18832 L .80708 .18792 L .80834 .18735 L .80954 .18666 L .81174 .18506 L .81417 .1827 L .81671 .17954 L .81909 .17585 L .82334 .1673 L .82793 .15485 L .83252 .13844 L .83754 .1151 L .84279 .08355 L .84767 .04628 L Mfstroke .84767 .04628 m .85252 0 L s .02381 .2 m .04235 .22868 L .06258 .25683 L .07241 .26843 L .08157 .27762 L .0904 .2848 L .09984 .2905 L .10474 .2926 L .10751 .29352 L .11007 .2942 L .1123 .29465 L .11357 .29485 L .11476 .295 L .1154 .29507 L .11601 .29512 L .11657 .29516 L .11716 .29519 L .11749 .29521 L .1178 .29522 L .11839 .29523 L .11872 .29524 L .11908 .29524 L .11942 .29524 L .11973 .29523 L .12034 .29522 L .12091 .29519 L .12143 .29516 L .12199 .29513 L .12325 .29501 L .12441 .29487 L .12652 .29452 L .12877 .29402 L .13348 .29255 L .13869 .29028 L .14798 .2846 L .15799 .27626 L .17773 .254 L .19675 .22716 L .21504 .19881 L .23451 .16881 L .25326 .14291 L .26281 .13165 L .27317 .12137 L .28316 .11362 L .28792 .11075 L .29237 .10855 L .29671 .10688 L .29915 .10615 L .30137 .10562 L .3038 .10519 L Mistroke .30518 .10501 L .30584 .10494 L .30645 .10488 L .30704 .10484 L .30768 .10481 L .30826 .10478 L .30879 .10477 L .30937 .10476 L .31 .10476 L .31032 .10477 L .31066 .10478 L .31128 .1048 L .31187 .10483 L .31251 .10488 L .31382 .105 L .31504 .10516 L .31618 .10533 L .31874 .10586 L .32149 .10661 L .32654 .10849 L .33116 .11076 L .34026 .11674 L .3502 .12541 L .3697 .14794 L .38846 .17471 L .40841 .20572 L .42762 .23507 L .44611 .26003 L .45548 .2707 L .46578 .28047 L .4705 .28419 L .47557 .28761 L .48032 .29026 L .48472 .29223 L .48959 .29384 L .49217 .29444 L .49348 .29469 L .4949 .2949 L .49613 .29504 L .49667 .29509 L .49727 .29514 L .49779 .29517 L .49835 .2952 L .49894 .29522 L .49949 .29523 L .49983 .29524 L .50014 .29524 L .50049 .29524 L .50085 .29523 L 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.18645 .11459 L .19104 .11164 L .19596 .10903 L .20089 .107 L .20539 .10569 L .20755 .10524 L .20869 .10504 L .20991 .10488 L .21096 .10476 L .21152 .10471 L .21213 .10467 L .21268 .10464 L .21319 .10461 L .21368 .1046 L .21419 .10459 L .21478 .10459 L .21511 .10459 L .21542 .10459 L .21599 .10461 L .21659 .10463 L .21712 .10466 L .21763 .10469 L .21877 .10479 L .22 .10494 L .22113 .1051 L .22366 .10559 L .22583 .10613 L Mistroke .22818 .10685 L .23248 .1085 L .24194 .11363 L .25201 .12112 L .272 .14072 L .29125 .16201 L .30009 .17098 L .30511 .17544 L .30978 .179 L .31442 .18181 L .31701 .18301 L .31937 .18384 L .32067 .18418 L .32187 .18442 L .32249 .18451 L .32318 .18458 L .32352 .18461 L .3239 .18463 L .32425 .18465 L .32457 .18465 L .32517 .18465 L .32572 .18463 L .32632 .18458 L .32666 .18454 L .32697 .1845 L .32759 .1844 L .32827 .18426 L .32949 .18393 L .33063 .18353 L .33185 .18299 L .33405 .18172 L .33666 .17969 L .33905 .1773 L .34424 .17015 L .34916 .16055 L .35378 .14865 L .3587 .13249 L .36754 .0927 L .37203 .06625 L .37687 .03222 L Mfstroke .37687 .03222 m .38047 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14265 L .16824 .13093 L .17746 .12179 L .18645 .11459 L .19104 .11164 L .19596 .10903 L .20089 .107 L .20539 .10569 L .20755 .10524 L .20869 .10504 L .20991 .10488 L .21096 .10476 L .21152 .10471 L .21213 .10467 L .21268 .10464 L .21319 .10461 L .21368 .1046 L .21419 .10459 L .21478 .10459 L .21511 .10459 L .21542 .10459 L .21599 .10461 L .21659 .10463 L .21712 .10466 L .21763 .10469 L .21877 .10479 L .22 .10494 L .22113 .1051 L .22366 .10559 L .22583 .10613 L Mistroke .22818 .10685 L .23248 .1085 L .24194 .11363 L .25201 .12112 L .272 .14072 L .29125 .16201 L .30009 .17098 L .30511 .17544 L .30978 .179 L .31442 .18181 L .31701 .18301 L .31937 .18384 L .32067 .18418 L .32187 .18442 L .32249 .18451 L .32318 .18458 L .32352 .18461 L .3239 .18463 L .32425 .18465 L .32457 .18465 L .32517 .18465 L .32572 .18463 L .32632 .18458 L .32666 .18454 L .32697 .1845 L .32759 .1844 L .32827 .18426 L .32949 .18393 L .33063 .18353 L .33185 .18299 L .33405 .18172 L .33666 .17969 L .33905 .1773 L .34424 .17015 L .34916 .16055 L .35378 .14865 L .3587 .13249 L .36754 .0927 L .37203 .06625 L .37687 .03222 L Mfstroke .37687 .03222 m .38047 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20323 .10635 L .20539 .10579 L .20755 .10535 L .20867 .10518 L .20991 .10502 L .21098 .10491 L .21198 .10484 L .21253 .10481 L .21304 .10479 L .2136 .10478 L .21419 .10477 L .21478 .10477 L .21511 .10478 L .21542 .10479 L .21599 .10481 L .21659 .10484 L .21763 .10492 L .21877 .10504 L .21999 .1052 L .2213 .10541 L .22366 .10592 L .22827 .10731 L .23248 .10905 L Mistroke .24161 .11433 L .25156 .12232 L .2711 .14396 L .28991 .17071 L .30989 .20328 L .32915 .23694 L .34768 .27105 L .36738 .31152 L .3772 .33538 L .38636 .36166 L .39574 .39466 L Mfstroke .39574 .39466 m .39694 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20323 .10635 L .20539 .10579 L .20755 .10535 L .20867 .10518 L .20991 .10502 L .21098 .10491 L .21198 .10484 L .21253 .10481 L .21304 .10479 L .2136 .10478 L .21419 .10477 L .21478 .10477 L .21511 .10478 L .21542 .10479 L .21599 .10481 L .21659 .10484 L .21763 .10492 L .21877 .10504 L .21999 .1052 L .2213 .10541 L .22366 .10592 L .22827 .10731 L .23248 .10905 L Mistroke .24161 .11433 L .25156 .12232 L .2711 .14396 L .28991 .17071 L .30989 .20328 L .32915 .23694 L .34768 .27105 L .36738 .31152 L .3772 .33538 L .38636 .36166 L .39574 .39466 L Mfstroke .39574 .39466 m .39694 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .10479 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10476 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .14358 L .28991 .16964 L .30989 .20032 L .32915 .22957 L .34768 .25423 L .35708 .2644 L .36205 .26894 L .36738 .27306 L .37252 .27621 L .3772 .27831 L .37946 .27904 L .38065 .27935 L .38191 .27961 L .38301 .27979 L .3836 .27986 L .38422 .27993 L .3848 .27997 L .38533 .28 L .38583 .28002 L .38636 .28002 L .38696 .28001 L .38729 .28 L .3876 .27998 L .38817 .27994 L .38877 .27988 L .38981 .27974 L .39096 .27952 L .39218 .27923 L .39349 .27882 L .39586 .27787 L .39822 .27663 L .40047 .27516 L .40468 .2716 L .40931 .26637 L .41422 .25916 L .42306 .24126 L .42755 .22935 L .4324 .21406 L .44117 .17879 L .45077 .12641 L .4609 .0511 L Mfstroke .4609 .0511 m .466 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .10479 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10476 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .14358 L .28991 .16964 L .30989 .20032 L .32915 .22957 L .34768 .25423 L .35708 .2644 L .36205 .26894 L .36738 .27306 L .37252 .27621 L .3772 .27831 L .37946 .27904 L .38065 .27935 L .38191 .27961 L .38301 .27979 L .3836 .27986 L .38422 .27993 L .3848 .27997 L .38533 .28 L .38583 .28002 L .38636 .28002 L .38696 .28001 L .38729 .28 L .3876 .27998 L .38817 .27994 L .38877 .27988 L .38981 .27974 L .39096 .27952 L .39218 .27923 L .39349 .27882 L .39586 .27787 L .39822 .27663 L .40047 .27516 L .40468 .2716 L .40931 .26637 L .41422 .25916 L .42306 .24126 L .42755 .22935 L .4324 .21406 L .44117 .17879 L .45077 .12641 L .4609 .0511 L Mfstroke .4609 .0511 m .466 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20059 L .32915 .23035 L .34768 .25623 L .35708 .26756 L .36738 .2782 L .37732 .28647 L .38636 .29218 L .39118 .2945 L .39575 .29623 L .40056 .29758 L .40328 .29813 L .40459 .29835 L .40579 .29851 L .40686 .29864 L .40802 .29875 L .40919 .29883 L .40984 .29887 L .41042 .2989 L .41098 .29892 L .41151 .29893 L .41207 .29894 L .41268 .29895 L .41321 .29895 L .41376 .29894 L .41424 .29893 L .41477 .29892 L .41543 .29889 L .41606 .29886 L .41724 .29879 L .4185 .29869 L .41988 .29856 L .42466 .29791 L .42917 .29709 L .43414 .29603 L .4394 .29485 L .44422 .29386 L .44656 .29345 L .44874 .29314 L .44993 .293 L .45101 .29289 L .45221 .2928 L .45287 .29277 L .4535 .29275 L .45408 .29273 L .4544 .29273 L .45471 .29273 L Mistroke .45533 .29274 L .45568 .29274 L .456 .29275 L .45656 .29278 L .45718 .29282 L .45776 .29287 L .4583 .29293 L .45962 .29311 L .46082 .29333 L .4622 .29365 L .46351 .29404 L .46574 .29488 L .4682 .29612 L .47076 .29779 L .47315 .29976 L .47743 .30438 L .48202 .31119 L .48691 .32102 L .49222 .33534 L .49724 .35308 L .50188 .37384 L Mfstroke .50188 .37384 m .50616 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20059 L .32915 .23035 L .34768 .25623 L .35708 .26756 L .36738 .2782 L .37732 .28647 L .38636 .29218 L .39118 .2945 L .39575 .29623 L .40056 .29758 L .40328 .29813 L .40459 .29835 L .40579 .29851 L .40686 .29864 L .40802 .29875 L .40919 .29883 L .40984 .29887 L .41042 .2989 L .41098 .29892 L .41151 .29893 L .41207 .29894 L .41268 .29895 L .41321 .29895 L .41376 .29894 L .41424 .29893 L .41477 .29892 L .41543 .29889 L .41606 .29886 L .41724 .29879 L .4185 .29869 L .41988 .29856 L .42466 .29791 L .42917 .29709 L .43414 .29603 L .4394 .29485 L .44422 .29386 L .44656 .29345 L .44874 .29314 L .44993 .293 L .45101 .29289 L .45221 .2928 L .45287 .29277 L .4535 .29275 L .45408 .29273 L .4544 .29273 L .45471 .29273 L Mistroke .45533 .29274 L .45568 .29274 L .456 .29275 L .45656 .29278 L .45718 .29282 L .45776 .29287 L .4583 .29293 L .45962 .29311 L .46082 .29333 L .4622 .29365 L .46351 .29404 L .46574 .29488 L .4682 .29612 L .47076 .29779 L .47315 .29976 L .47743 .30438 L .48202 .31119 L .48691 .32102 L .49222 .33534 L .49724 .35308 L .50188 .37384 L Mfstroke .50188 .37384 m .50616 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20057 L .32915 .23028 L .34768 .25604 L .35708 .26725 L .36738 .27766 L .3772 .28549 L .38159 .28827 L .38636 .29076 L .38898 .29189 L .39139 .29277 L .39366 .29346 L .39615 .29407 L .39723 .29428 L .39838 .29447 L .39938 .29461 L .40048 .29474 L .40164 .29483 L .40228 .29487 L .40287 .29489 L .40343 .29491 L .40393 .29491 L .40451 .29491 L .40506 .2949 L .40562 .29488 L .40622 .29485 L .40678 .29482 L .40729 .29478 L .40855 .29465 L .40974 .29449 L .41211 .29406 L .41469 .29341 L .41929 .29182 L .42355 .28984 L .4325 .28411 L .44238 .27534 L .46167 .25105 L .48022 .21864 L .49995 .17265 L .50979 .1436 L .51896 .1113 L .53723 .02368 L Mfstroke .53723 .02368 m .54071 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20057 L .32915 .23028 L .34768 .25604 L .35708 .26725 L .36738 .27766 L .3772 .28549 L .38159 .28827 L .38636 .29076 L .38898 .29189 L .39139 .29277 L .39366 .29346 L .39615 .29407 L .39723 .29428 L .39838 .29447 L .39938 .29461 L .40048 .29474 L .40164 .29483 L .40228 .29487 L .40287 .29489 L .40343 .29491 L .40393 .29491 L .40451 .29491 L .40506 .2949 L .40562 .29488 L .40622 .29485 L .40678 .29482 L .40729 .29478 L .40855 .29465 L .40974 .29449 L .41211 .29406 L .41469 .29341 L .41929 .29182 L .42355 .28984 L .4325 .28411 L .44238 .27534 L .46167 .25105 L .48022 .21864 L .49995 .17265 L .50979 .1436 L .51896 .1113 L .53723 .02368 L Mfstroke .53723 .02368 m .54071 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28838 L .38636 .2909 L .38898 .29204 L .39139 .29295 L .39366 .29366 L .39615 .2943 L .39838 .29473 L .3994 .29489 L .40048 .29502 L .40163 .29513 L .40214 .29517 L .40268 .29521 L .40327 .29524 L .40382 .29525 L .40416 .29526 L .40447 .29527 L .40506 .29527 L .40562 .29526 L .40622 .29524 L .40678 .29522 L .40729 .29519 L .40855 .29509 L .40974 .29496 L .41211 .29458 L .41469 .29401 L .41929 .29258 L .42355 .29078 L .4325 .28558 L .44238 .27771 L .46167 .25688 L .48022 .23201 L .49995 .20381 L .50979 .19052 L .51896 .17947 L .52779 .17074 L .53274 .16697 L .53723 .16441 L .53956 .16346 L .54073 .16309 L .54202 .16276 L Mistroke .54312 .16256 L .54373 .16247 L .5443 .16242 L .54482 .16238 L .54539 .16236 L .54593 .16236 L .54643 .16238 L .54706 .16242 L .54741 .16246 L .54774 .1625 L .54833 .16259 L .54897 .16272 L .55006 .163 L .55126 .16341 L .55255 .16398 L .55392 .16472 L .55641 .16647 L .55895 .16882 L .5613 .17155 L .56661 .17989 L .5713 .19009 L .57627 .20433 L .58077 .22071 L .58563 .24269 L .5944 .29622 L .599 .33285 L .60403 .38105 L Mfstroke .60403 .38105 m .60552 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28838 L .38636 .2909 L .38898 .29204 L .39139 .29295 L .39366 .29366 L .39615 .2943 L .39838 .29473 L .3994 .29489 L .40048 .29502 L .40163 .29513 L .40214 .29517 L .40268 .29521 L .40327 .29524 L .40382 .29525 L .40416 .29526 L .40447 .29527 L .40506 .29527 L .40562 .29526 L .40622 .29524 L .40678 .29522 L .40729 .29519 L .40855 .29509 L .40974 .29496 L .41211 .29458 L .41469 .29401 L .41929 .29258 L .42355 .29078 L .4325 .28558 L .44238 .27771 L .46167 .25688 L .48022 .23201 L .49995 .20381 L .50979 .19052 L .51896 .17947 L .52779 .17074 L .53274 .16697 L .53723 .16441 L .53956 .16346 L .54073 .16309 L .54202 .16276 L Mistroke .54312 .16256 L .54373 .16247 L .5443 .16242 L .54482 .16238 L .54539 .16236 L .54593 .16236 L .54643 .16238 L .54706 .16242 L .54741 .16246 L .54774 .1625 L .54833 .16259 L .54897 .16272 L .55006 .163 L .55126 .16341 L .55255 .16398 L .55392 .16472 L .55641 .16647 L .55895 .16882 L .5613 .17155 L .56661 .17989 L .5713 .19009 L .57627 .20433 L .58077 .22071 L .58563 .24269 L .5944 .29622 L .599 .33285 L .60403 .38105 L Mfstroke .60403 .38105 m .60552 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29486 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29523 L .40506 .29523 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29514 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29246 L .42473 .29011 L .434 .28436 L .44401 .27595 L .46374 .25355 L .48275 .22656 L .50102 .19794 L .52048 .16728 L .5392 .13983 L .55911 .11429 L .57828 .09257 L .59673 .0702 L .60626 .05515 L Mistroke .61113 .04566 L .61636 .03352 L .62126 .01982 L .62574 .00482 L Mfstroke .62574 .00482 m .62685 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29486 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29523 L .40506 .29523 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29514 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29246 L .42473 .29011 L .434 .28436 L .44401 .27595 L .46374 .25355 L .48275 .22656 L .50102 .19794 L .52048 .16728 L .5392 .13983 L .55911 .11429 L .57828 .09257 L .59673 .0702 L .60626 .05515 L Mistroke .61113 .04566 L .61636 .03352 L .62126 .01982 L .62574 .00482 L Mfstroke .62574 .00482 m .62685 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22675 L .50102 .19844 L .52048 .16856 L .5392 .14294 L .54875 .13192 L .55911 .12202 L .56395 .11822 L .56908 .11484 L Mistroke .57389 .1123 L .57828 .11054 L .5806 .10984 L .58281 .10932 L .58393 .10911 L .58513 .10894 L .58582 .10885 L .58645 .10879 L .58707 .10874 L .58767 .10871 L .58818 .10869 L .58874 .10868 L .58932 .10867 L .58987 .10868 L .59038 .1087 L .59084 .10872 L .59138 .10875 L .59189 .10879 L .59303 .10892 L .59408 .10907 L .59645 .10956 L .59868 .11019 L .60108 .11108 L .60542 .11323 L .61058 .11675 L .61531 .12094 L .6244 .13188 L .63434 .14879 L .64376 .1706 L .65382 .20175 L .66407 .24451 L .67375 .29862 L .68336 .37059 L Mfstroke .68336 .37059 m .68631 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22675 L .50102 .19844 L .52048 .16856 L .5392 .14294 L .54875 .13192 L .55911 .12202 L .56395 .11822 L .56908 .11484 L Mistroke .57389 .1123 L .57828 .11054 L .5806 .10984 L .58281 .10932 L .58393 .10911 L .58513 .10894 L .58582 .10885 L .58645 .10879 L .58707 .10874 L .58767 .10871 L .58818 .10869 L .58874 .10868 L .58932 .10867 L .58987 .10868 L .59038 .1087 L .59084 .10872 L .59138 .10875 L .59189 .10879 L .59303 .10892 L .59408 .10907 L .59645 .10956 L .59868 .11019 L .60108 .11108 L .60542 .11323 L .61058 .11675 L .61531 .12094 L .6244 .13188 L .63434 .14879 L .64376 .1706 L .65382 .20175 L .66407 .24451 L .67375 .29862 L .68336 .37059 L Mfstroke .68336 .37059 m .68631 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16843 L .5392 .14259 L .54875 .13136 L .55911 .12109 L .56908 .11334 L .57383 .11045 L Mistroke .57828 .10823 L .58262 .10653 L .58485 .10583 L .58728 .10522 L .58865 .10493 L .58993 .10471 L .59125 .10452 L .59245 .10439 L .59302 .10434 L .59363 .1043 L .59416 .10427 L .59474 .10424 L .59536 .10422 L .5957 .10421 L .59601 .10421 L .59658 .10421 L .59718 .10422 L .59777 .10424 L .5981 .10425 L .59841 .10426 L .5991 .1043 L .59972 .10435 L .60085 .10446 L .60207 .10462 L .60466 .10506 L .60738 .1057 L .612 .10717 L .61704 .10933 L .62646 .11475 L .63652 .12225 L .65645 .1402 L .66644 .14914 L .67083 .15266 L .67565 .15603 L .67789 .15737 L .68029 .15862 L .68237 .15951 L .68463 .16026 L .68595 .16059 L .68659 .16071 L .68717 .16081 L .68778 .16089 L .68835 .16094 L .68894 .16098 L .68928 .16099 L .68959 .16099 L .69014 .16098 L .69064 .16096 L .6912 .16091 L .6918 .16084 L Mistroke .69242 .16074 L .69302 .16061 L .69413 .16031 L .69527 .15991 L .69634 .15943 L .69878 .15798 L .70115 .15603 L .70371 .15329 L .70833 .14639 L .71256 .13754 L .71728 .12425 L .72167 .1081 L .72648 .0854 L .73161 .05435 L .73655 .01656 L Mfstroke .73655 .01656 m .73827 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16843 L .5392 .14259 L .54875 .13136 L .55911 .12109 L .56908 .11334 L .57383 .11045 L Mistroke .57828 .10823 L .58262 .10653 L .58485 .10583 L .58728 .10522 L .58865 .10493 L .58993 .10471 L .59125 .10452 L .59245 .10439 L .59302 .10434 L .59363 .1043 L .59416 .10427 L .59474 .10424 L .59536 .10422 L .5957 .10421 L .59601 .10421 L .59658 .10421 L .59718 .10422 L .59777 .10424 L .5981 .10425 L .59841 .10426 L .5991 .1043 L .59972 .10435 L .60085 .10446 L .60207 .10462 L .60466 .10506 L .60738 .1057 L .612 .10717 L .61704 .10933 L .62646 .11475 L .63652 .12225 L .65645 .1402 L .66644 .14914 L .67083 .15266 L .67565 .15603 L .67789 .15737 L .68029 .15862 L .68237 .15951 L .68463 .16026 L .68595 .16059 L .68659 .16071 L .68717 .16081 L .68778 .16089 L .68835 .16094 L .68894 .16098 L .68928 .16099 L .68959 .16099 L .69014 .16098 L .69064 .16096 L .6912 .16091 L .6918 .16084 L Mistroke .69242 .16074 L .69302 .16061 L .69413 .16031 L .69527 .15991 L .69634 .15943 L .69878 .15798 L .70115 .15603 L .70371 .15329 L .70833 .14639 L .71256 .13754 L .71728 .12425 L .72167 .1081 L .72648 .0854 L .73161 .05435 L .73655 .01656 L Mfstroke .73655 .01656 m .73827 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16844 L .5392 .14262 L .54875 .13142 L .55911 .12119 L .56908 .1135 L .57383 .11065 L Mistroke .57828 .10848 L .58262 .10685 L .58505 .10614 L .58728 .10562 L .58971 .1052 L .59108 .10503 L .59174 .10497 L .59236 .10492 L .59295 .10488 L .59325 .10486 L .59358 .10485 L .59416 .10483 L .59469 .10482 L .59531 .10482 L .59588 .10482 L .59651 .10484 L .59686 .10486 L .59718 .10487 L .59828 .10495 L .5989 .105 L .59949 .10507 L .60194 .10542 L .60406 .10586 L .60635 .10646 L .61155 .10832 L .61632 .11062 L .62541 .11657 L .63535 .12527 L .65482 .14824 L .67358 .17636 L .6935 .21106 L .71271 .24877 L .73118 .29139 L .74054 .31754 L .75083 .35254 L .76061 .39526 L Mfstroke .76061 .39526 m .76143 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16844 L .5392 .14262 L .54875 .13142 L .55911 .12119 L .56908 .1135 L .57383 .11065 L Mistroke .57828 .10848 L .58262 .10685 L .58505 .10614 L .58728 .10562 L .58971 .1052 L .59108 .10503 L .59174 .10497 L .59236 .10492 L .59295 .10488 L .59325 .10486 L .59358 .10485 L .59416 .10483 L .59469 .10482 L .59531 .10482 L .59588 .10482 L .59651 .10484 L .59686 .10486 L .59718 .10487 L .59828 .10495 L .5989 .105 L .59949 .10507 L .60194 .10542 L .60406 .10586 L .60635 .10646 L .61155 .10832 L .61632 .11062 L .62541 .11657 L .63535 .12527 L .65482 .14824 L .67358 .17636 L .6935 .21106 L .71271 .24877 L .73118 .29139 L .74054 .31754 L .75083 .35254 L .76061 .39526 L Mfstroke .76061 .39526 m .76143 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16844 L .5392 .14262 L .54875 .13141 L .55911 .12118 L .56908 .11349 L .57383 .11063 L Mistroke .57828 .10846 L .58262 .10682 L .58505 .1061 L .58728 .10558 L .58971 .10515 L .59108 .10498 L .59174 .10492 L .59236 .10487 L .59295 .10483 L .59325 .10481 L .59358 .10479 L .59416 .10477 L .59469 .10476 L .59531 .10476 L .59588 .10476 L .59651 .10478 L .59686 .10479 L .59718 .10481 L .59828 .10488 L .5989 .10493 L .59949 .10499 L .60194 .10534 L .60406 .10576 L .60635 .10635 L .61155 .10818 L .61632 .11044 L .62541 .11628 L .63535 .12481 L .65482 .14705 L .67358 .17349 L .6935 .20397 L .71271 .23221 L .73118 .25476 L .74073 .26335 L .74554 .26661 L .74825 .26808 L .75083 .26921 L .75317 .27 L .75449 .27034 L .75571 .27059 L .7563 .27068 L .75685 .27075 L .75735 .27081 L .75787 .27085 L .75845 .27088 L .75898 .27089 L .75955 .27089 L .76017 .27087 L .76049 .27085 L .76085 .27083 L Mistroke .76148 .27076 L .76211 .27068 L .76268 .27058 L .76398 .27029 L .76536 .26988 L .76767 .26893 L .7702 .26749 L .77269 .26565 L .77506 .26348 L .77951 .2582 L .7843 .25056 L .78953 .23954 L .79448 .22614 L .79897 .21108 L .80421 .18942 L .80904 .16495 L .81827 .10312 L .82313 .06082 L .82826 .00705 L Mfstroke .82826 .00705 m .82878 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16844 L .5392 .14262 L .54875 .13141 L .55911 .12118 L .56908 .11349 L .57383 .11063 L Mistroke .57828 .10846 L .58262 .10682 L .58505 .1061 L .58728 .10558 L .58971 .10515 L .59108 .10498 L .59174 .10492 L .59236 .10487 L .59295 .10483 L .59325 .10481 L .59358 .10479 L .59416 .10477 L .59469 .10476 L .59531 .10476 L .59588 .10476 L .59651 .10478 L .59686 .10479 L .59718 .10481 L .59828 .10488 L .5989 .10493 L .59949 .10499 L .60194 .10534 L .60406 .10576 L .60635 .10635 L .61155 .10818 L .61632 .11044 L .62541 .11628 L .63535 .12481 L .65482 .14705 L .67358 .17349 L .6935 .20397 L .71271 .23221 L .73118 .25476 L .74073 .26335 L .74554 .26661 L .74825 .26808 L .75083 .26921 L .75317 .27 L .75449 .27034 L .75571 .27059 L .7563 .27068 L .75685 .27075 L .75735 .27081 L .75787 .27085 L .75845 .27088 L .75898 .27089 L .75955 .27089 L .76017 .27087 L .76049 .27085 L .76085 .27083 L Mistroke .76148 .27076 L .76211 .27068 L .76268 .27058 L .76398 .27029 L .76536 .26988 L .76767 .26893 L .7702 .26749 L .77269 .26565 L .77506 .26348 L .77951 .2582 L .7843 .25056 L .78953 .23954 L .79448 .22614 L .79897 .21108 L .80421 .18942 L .80904 .16495 L .81827 .10312 L .82313 .06082 L .82826 .00705 L Mfstroke .82826 .00705 m .82878 0 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16844 L .5392 .14262 L .54875 .13141 L .55911 .12118 L .56908 .11349 L .57383 .11064 L Mistroke .57828 .10846 L .58262 .10682 L .58505 .1061 L .58728 .10558 L .58971 .10516 L .59108 .10499 L .59174 .10492 L .59236 .10487 L .59295 .10483 L .59325 .10481 L .59358 .1048 L .59416 .10478 L .59469 .10477 L .59531 .10476 L .59588 .10477 L .59651 .10478 L .59686 .1048 L .59718 .10481 L .59828 .10488 L .5989 .10494 L .59949 .105 L .60194 .10534 L .60406 .10577 L .60635 .10636 L .61155 .10819 L .61632 .11046 L .62541 .11631 L .63535 .12485 L .65482 .14718 L .67358 .17383 L .6935 .20484 L .71271 .23436 L .73118 .25979 L .74054 .27086 L .75083 .28129 L .76072 .28945 L .76975 .29526 L .77909 .2997 L .78424 .30154 L .78912 .30295 L .80749 .30667 L .81237 .30777 L .81696 .30913 L .8195 .31008 L .82225 .31133 L .82704 .31417 L .83162 .31794 L .83663 .32361 L .84187 .33179 L .84675 .34206 L Mistroke .85179 .35603 L .85655 .37316 L Mfstroke .85655 .37316 m .86181 .4 L s .02381 .29524 m .02437 .29523 L .02489 .29522 L .02548 .2952 L .02604 .29517 L .02704 .2951 L .02812 .295 L .02931 .29485 L .03057 .29465 L .03284 .29418 L .03518 .29357 L .03737 .29287 L .04235 .29082 L .04744 .28809 L .0521 .28506 L .06258 .27642 L .08164 .2551 L .09997 .22947 L .11948 .19932 L .13826 .17032 L .15822 .14266 L .16824 .13093 L .17746 .1218 L .18645 .11462 L .19104 .11168 L .19596 .10908 L .20089 .10708 L .20539 .10579 L .20755 .10535 L .20869 .10517 L .20991 .10501 L .21096 .1049 L .21152 .10486 L .21213 .10482 L .21268 .1048 L .21319 .10478 L .21368 .10477 L .21419 .10476 L .21478 .10477 L .21511 .10477 L .21542 .10478 L .21599 .1048 L .21659 .10483 L .21763 .10491 L .21877 .10502 L .21999 .10518 L .2213 .1054 L .22366 .1059 L .22827 .10728 L .23248 .10902 L Mistroke .24161 .11427 L .25156 .1222 L .2711 .1436 L .28991 .16972 L .30989 .20058 L .32915 .23029 L .34768 .25605 L .35708 .26727 L .36738 .27771 L .3772 .28557 L .38159 .28837 L .38636 .29089 L .38898 .29203 L .39139 .29293 L .39366 .29364 L .39615 .29428 L .39838 .29471 L .3994 .29487 L .40048 .295 L .40103 .29506 L .40163 .29511 L .40218 .29515 L .40268 .29518 L .40328 .29521 L .40361 .29522 L .40392 .29523 L .40447 .29524 L .40506 .29524 L .40566 .29523 L .4062 .29521 L .40682 .29518 L .40742 .29515 L .40847 .29506 L .40962 .29493 L .41088 .29475 L .41222 .29452 L .41462 .29398 L .41942 .29247 L .42473 .29012 L .434 .28437 L .44401 .27597 L .46374 .25362 L .48275 .22674 L .50102 .19839 L .52048 .16844 L .5392 .14262 L .54875 .13141 L .55911 .12118 L .56908 .11349 L .57383 .11064 L Mistroke .57828 .10846 L .58262 .10682 L .58505 .1061 L .58728 .10558 L .58971 .10516 L .59108 .10499 L .59174 .10492 L .59236 .10487 L .59295 .10483 L .59325 .10481 L .59358 .1048 L .59416 .10478 L .59469 .10477 L .59531 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07ooOol0Goo00<007ooOol0Goo00<007ooOol0 Goo00<007ooOol0Goo00<007ooOol07oo00<007ooOol07oo00<0 07ooOol0"], ImageRangeCache->{{{48.625, 485.063}, {583.25, 409.5}} -> {-2.72956, \ 7.99838, 0.031154, 0.0198333}}], Cell["\[FilledSmallSquare]", "Text", TextAlignment->Right], Cell["\<\ Zum Schluss wird die Taylor-Reihe des arctan hergeleitet. Damit ergibt sich \ eine neue M\[ODoubleDot]glichkeit \[Pi] numerisch zu berechnen.\ \>", "Text"], Cell[TextData[{ StyleBox["Satz 7.7:\n", FontFamily->"Arial", FontVariations->{"Underline"->True}], StyleBox["F\[UDoubleDot]r ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`x \[Element] \[DoubleStruckCapitalR]\)]], " mit ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\ x\ \[RightBracketingBar] \[LessEqual] 1\)]], " gilt ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(arctan( x)\ = \ \[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), "TraditionalForm"], " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(2 \[Nu]\ + \ 1\), FontSize->10]], \(\(x\^\(2 \[Nu] + 1\)\)\(\ \)\(.\)\)}], TraditionalForm]]], "\n\n", StyleBox["Beweis:", FontFamily->"Arial"], "\nIm Beweis wird hier \[ADoubleDot]hnlich verfahren wie bei der \ Entwicklung von ln(", StyleBox["x", FontSlant->"Italic"], " + 1) in Satz 7.5.\nF\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[Element] \[DoubleStruckCapitalR]\)]], " und ", Cell[BoxData[ \(TraditionalForm\`n \[Element] \[DoubleStruckCapitalN]\)]], " sei\n ", Cell[BoxData[ \(TraditionalForm\`\(\(\(f\_n\)(x)\)\(\ \)\)\)]], "\t", Cell[BoxData[ FormBox[ RowBox[{":=", " ", RowBox[{\(arctan(x)\), " ", "-", RowBox[{"(", RowBox[{ FractionBox[ StyleBox["x", FontSize->10], StyleBox["1", FontSize->10]], "-", FractionBox[ StyleBox[\(x\^3\), FontSize->10], StyleBox["3", FontSize->10]], "+", \(-\[CenterEllipsis]\), "+", RowBox[{\(\((\(-1\))\)\^\(n - 1\)\), " ", FractionBox[ StyleBox[\(x\^\(2 n - 1\)\), FontSize->10], StyleBox[\(2 n\ - \ 1\), FontSize->10]]}]}], ")"}]}]}], TraditionalForm]]], "\n \n\t\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox[":", FontColor->GrayLevel[1]], "="}]], " ", RowBox[{ FormBox[\(arctan( x)\ - \ \[CapitalSigma]\+\(\[Nu] = 0\)\%\(n - 1\)\), "TraditionalForm"], " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(2 \[Nu]\ + \ 1\), FontSize->10]], \(\(x\^\(2 \[Nu] + 1\)\)\(\ \)\(.\)\)}]}], TraditionalForm]]], "\n\t\t\nAlso ist ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\)(0) = 0\)]], " und\n\n\t", Cell[BoxData[ \(TraditionalForm\`\(\(\(f\_n\^\[Prime]\)(x)\)\(\ \)\)\)]], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox[":", FontColor->GrayLevel[1]], "="}]], " ", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\^2\), FontSize->10]], "-", RowBox[{ StyleBox["(", FontSize->14], StyleBox[" ", FontSize->14], \(1 - x\^2 + x\^4 - \(+\[CenterEllipsis]\) + \((\(-1\))\)\^\(n - \ 1\)\ \ x\^\(2 n - 2\)\), StyleBox[" ", FontSize->14], StyleBox[")", FontSize->14]}]}]}], TraditionalForm]]], "\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox[":", FontColor->GrayLevel[1]], "="}]], " ", RowBox[{ FormBox[ RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\^2\), FontSize->10]], "-", " ", \(\[CapitalSigma]\+\(\[Nu] = 0\)\%\(n - 1\)\)}], "TraditionalForm"], " ", \(\((\(-1\))\)\^\[Nu]\), \(x\^\(2 \[Nu]\)\), " "}]}], TraditionalForm]]], "\n\t\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox[":", FontColor->GrayLevel[1]], "="}]], " ", FormBox[ RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\^2\), FontSize->10]], "-", RowBox[{"(", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox[\(1\ + \ x\^2\), FontSize->10]], "-", FractionBox[ StyleBox[\(\(\((\(-1\))\)\^n\) x\^\(2 n\)\), FontSize->10], StyleBox[\(1\ + \ x\^2\), FontSize->10]]}], " ", ")"}]}], "TraditionalForm"]}], TraditionalForm]]], "\n\t\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox[":", FontColor->GrayLevel[1]], "="}]], " ", FormBox[ RowBox[{ FractionBox[ StyleBox[\(\((\(-x\^2\))\)\^n\), FontSize->10], StyleBox[\(1\ + \ x\^2\), FontSize->10]], " ", "."}], "TraditionalForm"]}], TraditionalForm]]], "\n\nEs sei nun ", StyleBox["n", FontSlant->"Italic"], " gerade, etwa ", Cell[BoxData[ \(TraditionalForm\`n = 2 \( \(m\)\(\ \)\(.\)\)\)]], " Dann ist ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\^\[Prime]\)(x)\ \[GreaterEqual] 0\)]], " f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[GreaterEqual] 0\)]], " und damit ", Cell[BoxData[ \(TraditionalForm\`f\_n\)]], " f\[UDoubleDot]r diese ", StyleBox["x", FontSlant->"Italic"], " monoton wachsend. Da ", Cell[BoxData[ \(TraditionalForm\`\(\(\(f\_n\)(0) = 0\)\(\ \)\(,\)\)\)]], " gilt ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\)(x)\ \[GreaterEqual] 0\)]], " f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`\(\(x \[GreaterEqual] 0\)\(\ \ \)\(,\)\)\)]], "\n\nd.h.\t\t", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(arctan( x)\ \[GreaterEqual] \ \ \[CapitalSigma]\+\(\[Nu] = \ 0\)\%\(2 m - 1\)\), "TraditionalForm"], " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(2 \[Nu]\ + \ 1\), FontSize->10]], " ", \(\(x\^\(2 \[Nu] + 1\)\)\(\ \)\(.\)\)}]}], TraditionalForm]]], "\n\nNun sei ", StyleBox["n", FontSlant->"Italic"], " ungerade, etwa ", Cell[BoxData[ \(TraditionalForm\`n = 2 m + \(\(1\)\(\ \)\(.\)\)\)]], " Dann ist ", Cell[BoxData[ \(TraditionalForm\`f\_n' \((x)\)\ \[LessEqual] \ 0\)]], " f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[GreaterEqual] 0\)]], " und analog zum vorigen Ergebnis folgt \n\t\t", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(arctan( x)\ \[LessEqual] \ \ \ \[CapitalSigma]\+\(\[Nu] = 0\)\%\(2 \ m\)\), "TraditionalForm"], " "}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(2 \[Nu]\ + \ 1\), FontSize->10]], " ", \(\(x\^\(2 \[Nu] + 1\)\)\(\ \)\(.\)\)}]}], TraditionalForm]]], "\n\nAlso gilt f\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`x \[GreaterEqual] 0\)]], "\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[CapitalSigma]\+\(\[Nu] = 0\)\%\(2 m - 1\)\)\), "TraditionalForm"], " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], RowBox[{ RowBox[{"2", StyleBox["\[Nu]", FontSize->10]}], StyleBox[" ", FontSize->10], StyleBox["+", FontSize->10], StyleBox[" ", FontSize->10], StyleBox["1", FontSize->10]}]], \(x\^\(2 \[Nu] + 1\)\)}], "\[LessEqual]", " ", RowBox[{ FormBox[\(arctan( x)\ \[LessEqual] \ \ \[CapitalSigma]\+\(\[Nu] = 0\)\%\(2 \ m\)\), "TraditionalForm"], " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], RowBox[{ RowBox[{"2", StyleBox["\[Nu]", FontSize->10]}], StyleBox[" ", FontSize->10], StyleBox["+", FontSize->10], StyleBox[" ", FontSize->10], StyleBox["1", FontSize->10]}]], \(\(x\^\(2 \[Nu] + 1\)\)\(\ \)\(.\)\)}]}], TraditionalForm]]], "\n\nDie Differenz zwischen der rechten und linken Seite \ betr\[ADoubleDot]gt ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(x\^\(4 m + 1\)\), FontSize->10], StyleBox[\(4 m\ + \ 1\), FontSize->10]], TraditionalForm]]], ". \nGilt nun noch ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] x \[LessEqual] 1\)]], ", so folgt mit ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\(\(lim\)\(\ \)\)\+\(m \[Rule] \[Infinity]\)\), RowBox[{"(", FractionBox[ StyleBox[\(x\^\(4 m + 1\)\), FontSize->10], StyleBox[\(4 m\ + \ 1\), FontSize->10]], ")"}]}], "=", "0"}], TraditionalForm]]], "\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(arctan( x)\ = \ \ \[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), "TraditionalForm"], " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], RowBox[{ RowBox[{"2", StyleBox["\[Nu]", FontSize->10]}], StyleBox[" ", FontSize->10], StyleBox["+", FontSize->10], StyleBox[" ", FontSize->10], StyleBox["1", FontSize->10]}]], \(\(x\^\(2 \[Nu] + 1\)\)\(\ \)\(.\)\)}]}], TraditionalForm]]], "\n\t\t\nNun ist arctan eine ungerade Funktion und auch ", Cell[BoxData[ \(TraditionalForm\`f\_n\)]], " ist ungerade, d.h. ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\)(x) = \(-\(\(f\_n\)(\(-x\))\)\)\)]], " . F\[UDoubleDot]r ", Cell[BoxData[ \(TraditionalForm\`\(-1\) \[LessEqual] x \[LessEqual] 0\)]], " gilt ", Cell[BoxData[ \(TraditionalForm\`\(\(\(lim\)\(\ \)\)\+\(n \[Rule] \[Infinity]\)\) \((\ \(f\_n\)(\(-x\)))\) = 0\)]], " und damit gilt auch ", Cell[BoxData[ \(TraditionalForm\`\(\(\(lim\)\(\ \)\)\+\(n \[Rule] \[Infinity]\)\) \((\ \(f\_n\)(x))\) = \(\(0\)\(\ \ \)\(.\)\)\)]], "\n\nBeide Ergebnisse zusammen bedeuten:\n\n", StyleBox["F\[UDoubleDot]r ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`x\)]], " mit ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(\ \)\(x\)\(\ \)\(|\)\(\(\[LessEqual]\)\(\ \ \)\(1\)\)\)\)]], " gilt ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(arctan( x)\ = \ \[CapitalSigma]\+\(\[Nu] = 0\)\%\[Infinity]\), "TraditionalForm"], " ", FractionBox[ StyleBox[\(\((\(-1\))\)\^\[Nu]\), FontSize->10], StyleBox[\(2 \[Nu]\ + \ 1\), FontSize->10]], " ", \(\(x\^\(2 \[Nu] + 1\)\)\(\ \)\(.\)\)}], TraditionalForm]]], "\n" }], "Text"], 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.39332 L .17706 .37509 L .1955 .36151 L .20483 .35604 L .21512 .35107 L .22487 .34733 L .234 .34467 L .23899 .34355 L .24372 .34269 L .24797 .3421 L .2502 .34185 L .25262 .34164 L .25394 .34154 L .25517 .34146 L .25631 .3414 L .25757 .34135 L .2587 .34131 L .25924 .3413 L .25975 .34129 L .26033 .34128 L .26086 .34127 L .26144 .34127 L .26206 .34127 L .26268 .34127 L .26326 .34128 L .2636 .34128 L .2639 .34129 L .26459 .3413 L .26522 .34132 L .26591 .34134 L .26729 .34139 L .26979 .34153 L .27213 .3417 L .27718 .34223 L .28194 .34291 L .29087 .34465 L .30047 .34718 L .30933 .35009 L .32824 .35804 L .34643 .36773 L .36579 .38001 L .38443 .39351 L .40424 .4094 L Mistroke .42333 .42598 L .44168 .44285 L .46122 .46156 L .48003 .48007 L .50001 .50001 L .51927 .51923 L .5378 .53748 L .5575 .55639 L .57648 .57385 L .59473 .58974 L .61416 .60541 L .63286 .61907 L .65274 .63179 L .67189 .64203 L .69032 .64978 L .69983 .65291 L .70992 .65553 L .71443 .65646 L .71928 .65728 L .72387 .6579 L .72806 .65831 L .73035 .65848 L .7316 .65855 L .73278 .65861 L .73388 .65866 L .73446 .65868 L .73508 .65869 L .7356 .6587 L .73616 .65871 L .73668 .65872 L .73716 .65873 L .73778 .65873 L .73813 .65873 L .73845 .65873 L .73904 .65873 L .73967 .65872 L .74023 .65871 L .74076 .6587 L .74196 .65867 L .74325 .65862 L .74444 .65856 L .74711 .65838 L .74938 .65819 L .75181 .65793 L .75618 .65732 L .76138 .65638 L .76611 .6553 L .77531 .65261 L .78529 .64876 L .80491 .63822 L Mistroke .82381 .62417 L .84198 .60681 L .86133 .58394 L .87995 .55743 L .89975 .52414 L .91882 .48685 L .93716 .44592 L .95668 .39665 L .97547 .34343 L .97619 .34127 L Mfstroke .02381 .65873 m .04235 .60595 L .06258 .5547 L .08157 .51233 L .09984 .47661 L .11928 .44376 L .138 .41693 L .1579 .39332 L .17706 .37509 L .1955 .36151 L .20483 .35604 L .21512 .35107 L .22487 .34733 L .234 .34467 L .23899 .34355 L .24372 .34269 L .24797 .3421 L .2502 .34185 L .25262 .34164 L .25394 .34154 L .25517 .34146 L .25631 .3414 L .25757 .34135 L .2587 .34131 L .25924 .3413 L .25975 .34129 L .26033 .34128 L .26086 .34127 L .26144 .34127 L .26206 .34127 L .26268 .34127 L .26326 .34128 L .2636 .34128 L .2639 .34129 L .26459 .3413 L .26522 .34132 L .26591 .34134 L .26729 .34139 L .26979 .34153 L .27213 .3417 L .27718 .34223 L .28194 .34291 L .29087 .34465 L .30047 .34718 L .30933 .35009 L .32824 .35804 L .34643 .36773 L .36579 .38001 L .38443 .39351 L .40424 .4094 L Mistroke .42333 .42598 L .44168 .44285 L .46122 .46156 L .48003 .48007 L .50001 .50001 L .51927 .51923 L .5378 .53748 L .5575 .55639 L .57648 .57385 L .59473 .58974 L .61416 .60541 L .63286 .61907 L .65274 .63179 L .67189 .64203 L .69032 .64978 L .69983 .65291 L .70992 .65553 L .71443 .65646 L .71928 .65728 L .72387 .6579 L .72806 .65831 L .73035 .65848 L .7316 .65855 L .73278 .65861 L .73388 .65866 L .73446 .65868 L .73508 .65869 L .7356 .6587 L .73616 .65871 L .73668 .65872 L .73716 .65873 L .73778 .65873 L .73813 .65873 L .73845 .65873 L .73904 .65873 L .73967 .65872 L .74023 .65871 L .74076 .6587 L .74196 .65867 L .74325 .65862 L .74444 .65856 L .74711 .65838 L .74938 .65819 L .75181 .65793 L .75618 .65732 L .76138 .65638 L .76611 .6553 L .77531 .65261 L .78529 .64876 L .80491 .63822 L Mistroke .82381 .62417 L .84198 .60681 L .86133 .58394 L .87995 .55743 L .89975 .52414 L .91882 .48685 L .93716 .44592 L .95668 .39665 L .97547 .34343 L .97619 .34127 L Mfstroke .13131 0 m .138 .03007 L .1579 .1017 L .17706 .15651 L .1955 .1986 L .21512 .23429 L .234 .2618 L .25217 .28349 L .2715 .30289 L .29011 .31913 L .3099 .33485 L .32896 .34927 L .34729 .36307 L .3668 .37809 L .38559 .39317 L .40555 .41003 L .42478 .42713 L .44328 .44432 L .46296 .46326 L .48192 .48195 L .50014 .50014 L .51955 .5195 L .53822 .5379 L .55808 .55697 L .5772 .57467 L .5956 .59096 L .61518 .60745 L .63402 .62256 L .65215 .63651 L .67144 .65103 L .69001 .66509 L .70976 .68076 L .72878 .69737 L .74707 .71569 L .76654 .73892 L .78528 .76636 L .8052 .80284 L .82439 .84723 L .84286 .90072 L .8625 .97195 L s .8625 .97195 m .8687 1 L s .13131 0 m .138 .03007 L .1579 .1017 L .17706 .15651 L .1955 .1986 L .21512 .23429 L .234 .2618 L .25217 .28349 L .2715 .30289 L .29011 .31913 L .3099 .33485 L .32896 .34927 L .34729 .36307 L .3668 .37809 L .38559 .39317 L .40555 .41003 L .42478 .42713 L .44328 .44432 L .46296 .46326 L .48192 .48195 L .50014 .50014 L .51955 .5195 L .53822 .5379 L .55808 .55697 L .5772 .57467 L .5956 .59096 L .61518 .60745 L .63402 .62256 L .65215 .63651 L .67144 .65103 L .69001 .66509 L .70976 .68076 L .72878 .69737 L .74707 .71569 L .76654 .73892 L .78528 .76636 L .8052 .80284 L .82439 .84723 L .84286 .90072 L .8625 .97195 L s .8625 .97195 m .8687 1 L s .1089 1 m .11928 .85506 L .138 .66884 L .14823 .59163 L .1579 .53174 L .17633 .44644 L .18586 .4145 L .19595 .38788 L .20533 .36879 L .21411 .35505 L .22383 .34378 L .23272 .33651 L .23745 .33366 L .24183 .33157 L .24665 .32984 L .24934 .32911 L .25178 .32859 L .25412 .3282 L .2554 .32804 L .25659 .32791 L .2577 .32782 L .2587 .32775 L .25924 .32773 L .25982 .3277 L .26044 .32768 L .26102 .32767 L .26164 .32767 L .26222 .32767 L .26255 .32767 L .26285 .32767 L .26319 .32768 L .26354 .32769 L .2642 .32771 L .26481 .32773 L .26621 .32782 L .26743 .32791 L .26853 .32802 L .27101 .32832 L .2758 .32914 L .28022 .33015 L .2902 .33323 L .30912 .34148 L .3273 .35162 L .34667 .36415 L .3653 .37754 L .38512 .39299 L .4042 .40893 L .42256 .42513 L .44209 .4432 L .4609 .46125 L Mistroke .48089 .48093 L .50014 .50014 L .51867 .51863 L .53838 .53805 L .55736 .55628 L .57561 .57321 L .59504 .59042 L .61374 .60608 L .63362 .62164 L .65277 .63546 L .67119 .64746 L .69079 .65847 L .70053 .66307 L .70966 .66672 L .7184 .66949 L .72331 .67068 L .72781 .67151 L .73009 .67183 L .73135 .67197 L .73253 .67209 L .73363 .67217 L .73483 .67225 L .73535 .67227 L .73591 .6723 L .73639 .67231 L .73691 .67232 L .73723 .67233 L .73758 .67233 L .73821 .67234 L .73882 .67233 L .73939 .67232 L .74 .6723 L .74068 .67228 L .74139 .67224 L .74205 .6722 L .74326 .6721 L .74436 .67199 L .74685 .67165 L .74912 .67123 L .75155 .67066 L .75593 .6693 L .76113 .66707 L .76586 .66441 L .77506 .65727 L .78025 .65195 L .78503 .6461 L .79403 .63235 L .80393 .61239 L .81326 .58812 L .82328 .555 L Mistroke .83275 .51582 L .84308 .46276 L .86215 .32993 L .8716 .24364 L .88167 .13368 L Mfstroke .88167 .13368 m .89105 0 L s .1089 1 m .11928 .85506 L .138 .66884 L .14823 .59163 L .1579 .53174 L .17633 .44644 L .18586 .4145 L .19595 .38788 L .20533 .36879 L .21411 .35505 L .22383 .34378 L .23272 .33651 L .23745 .33366 L .24183 .33157 L .24665 .32984 L .24934 .32911 L .25178 .32859 L .25412 .3282 L .2554 .32804 L .25659 .32791 L .2577 .32782 L .2587 .32775 L .25924 .32773 L .25982 .3277 L .26044 .32768 L .26102 .32767 L .26164 .32767 L .26222 .32767 L .26255 .32767 L .26285 .32767 L .26319 .32768 L .26354 .32769 L .2642 .32771 L .26481 .32773 L .26621 .32782 L .26743 .32791 L .26853 .32802 L .27101 .32832 L .2758 .32914 L .28022 .33015 L .2902 .33323 L .30912 .34148 L .3273 .35162 L .34667 .36415 L .3653 .37754 L .38512 .39299 L .4042 .40893 L .42256 .42513 L .44209 .4432 L .4609 .46125 L Mistroke .48089 .48093 L .50014 .50014 L .51867 .51863 L .53838 .53805 L .55736 .55628 L .57561 .57321 L .59504 .59042 L .61374 .60608 L .63362 .62164 L .65277 .63546 L .67119 .64746 L .69079 .65847 L .70053 .66307 L .70966 .66672 L .7184 .66949 L .72331 .67068 L .72781 .67151 L .73009 .67183 L .73135 .67197 L .73253 .67209 L .73363 .67217 L .73483 .67225 L .73535 .67227 L .73591 .6723 L .73639 .67231 L .73691 .67232 L .73723 .67233 L .73758 .67233 L .73821 .67234 L .73882 .67233 L .73939 .67232 L .74 .6723 L .74068 .67228 L .74139 .67224 L .74205 .6722 L .74326 .6721 L .74436 .67199 L .74685 .67165 L .74912 .67123 L .75155 .67066 L .75593 .6693 L .76113 .66707 L .76586 .66441 L .77506 .65727 L .78025 .65195 L .78503 .6461 L .79403 .63235 L .80393 .61239 L .81326 .58812 L .82328 .555 L Mistroke .83275 .51582 L .84308 .46276 L .86215 .32993 L .8716 .24364 L .88167 .13368 L Mfstroke .88167 .13368 m .89105 0 L s .17328 0 m .17539 .01951 L .18554 .09196 L .19518 .14526 L .20469 .18617 L .21351 .21602 L .22254 .24021 L .2323 .26083 L .24257 .27786 L .25226 .29068 L .27194 .3105 L .29089 .32526 L .30912 .33786 L .32852 .35098 L .3472 .36403 L .36705 .37872 L .38618 .39382 L .40458 .40924 L .42415 .42657 L .443 .44405 L .46112 .46146 L .48042 .48046 L .49899 .49899 L .51873 .5187 L .53775 .53744 L .55605 .55504 L .57551 .57313 L .59426 .58975 L .61418 .60647 L .63337 .6216 L .65183 .63527 L .67147 .64901 L .69038 .6618 L .70857 .67436 L .72793 .68939 L .74657 .70794 L .75673 .72113 L .76638 .73674 L .77602 .7564 L .78474 .77885 L .7942 .80969 L .80427 .85211 L .81353 .90229 L .82353 .97189 L s .82353 .97189 m .82668 1 L s .17328 0 m .17539 .01951 L .18554 .09196 L .19518 .14526 L .20469 .18617 L .21351 .21602 L .22254 .24021 L .2323 .26083 L .24257 .27786 L .25226 .29068 L .27194 .3105 L .29089 .32526 L .30912 .33786 L .32852 .35098 L .3472 .36403 L .36705 .37872 L .38618 .39382 L .40458 .40924 L .42415 .42657 L .443 .44405 L .46112 .46146 L .48042 .48046 L .49899 .49899 L .51873 .5187 L .53775 .53744 L .55605 .55504 L .57551 .57313 L .59426 .58975 L .61418 .60647 L .63337 .6216 L .65183 .63527 L .67147 .64901 L .69038 .6618 L .70857 .67436 L .72793 .68939 L .74657 .70794 L .75673 .72113 L .76638 .73674 L .77602 .7564 L .78474 .77885 L .7942 .80969 L .80427 .85211 L .81353 .90229 L .82353 .97189 L s .82353 .97189 m .82668 1 L s .15847 1 m .16637 .82239 L .17602 .66552 L .18552 .5538 L .19435 .47892 L .19942 .44573 L .2042 .41986 L .21313 .38298 L .21782 .36852 L .22277 .35621 L .22741 .34702 L .23163 .34033 L .23618 .33466 L .24106 .33009 L .24323 .32849 L .24558 .32702 L .24986 .32499 L .25219 .3242 L .25437 .32364 L .25562 .32339 L .25677 .32321 L .25803 .32305 L .25866 .32299 L .25935 .32294 L .25968 .32292 L .26003 .3229 L .26035 .32289 L .26068 .32287 L .2613 .32286 L .26188 .32285 L .2622 .32286 L .26254 .32286 L .26316 .32287 L .26354 .32289 L .2639 .3229 L .26458 .32294 L .26578 .32304 L .2669 .32315 L .26944 .32351 L .27188 .32396 L .27452 .32458 L .27931 .32596 L .28824 .3293 L .30795 .33911 L .32693 .35054 L .34518 .36278 L .36461 .3769 L .38331 .3915 L .40319 .40805 L .42234 .42493 L Mistroke .44076 .44194 L .46036 .46072 L .47924 .47929 L .49738 .49738 L .5167 .51668 L .5353 .53504 L .55507 .55412 L .57412 .57185 L .59243 .58817 L .61193 .60463 L .63069 .61952 L .65064 .63428 L .66986 .64738 L .68834 .65878 L .69772 .66398 L .70801 .66906 L .71791 .6731 L .72224 .67453 L .72695 .67578 L .72944 .6763 L .73175 .67668 L .73396 .67694 L .73507 .67703 L .73571 .67708 L .7363 .67711 L .73689 .67713 L .73722 .67714 L .73753 .67714 L .73815 .67715 L .7385 .67714 L .73882 .67714 L .73939 .67712 L .74001 .6771 L .74059 .67706 L .74113 .67703 L .74233 .67691 L .74366 .67673 L .74504 .67648 L .74634 .67619 L .74872 .67551 L .75132 .67452 L .75362 .67342 L .75606 .67198 L .76068 .66845 L .76496 .66405 L .7699 .65742 L .77524 .64799 L .78029 .63653 L .78493 .62338 L .79397 .58884 L Mistroke .7991 .5628 L .80389 .53334 L .81367 .45412 L .82258 .35344 L .83174 .21233 L .84171 .00215 L Mfstroke .84171 .00215 m .84179 0 L s .15847 1 m .16637 .82239 L .17602 .66552 L .18552 .5538 L .19435 .47892 L .19942 .44573 L .2042 .41986 L .21313 .38298 L .21782 .36852 L .22277 .35621 L .22741 .34702 L .23163 .34033 L .23618 .33466 L .24106 .33009 L .24323 .32849 L .24558 .32702 L .24986 .32499 L .25219 .3242 L .25437 .32364 L .25562 .32339 L .25677 .32321 L .25803 .32305 L .25866 .32299 L .25935 .32294 L .25968 .32292 L .26003 .3229 L .26035 .32289 L .26068 .32287 L .2613 .32286 L .26188 .32285 L .2622 .32286 L .26254 .32286 L .26316 .32287 L .26354 .32289 L .2639 .3229 L .26458 .32294 L .26578 .32304 L .2669 .32315 L .26944 .32351 L .27188 .32396 L .27452 .32458 L .27931 .32596 L .28824 .3293 L .30795 .33911 L .32693 .35054 L .34518 .36278 L .36461 .3769 L .38331 .3915 L .40319 .40805 L .42234 .42493 L Mistroke .44076 .44194 L .46036 .46072 L .47924 .47929 L .49738 .49738 L .5167 .51668 L .5353 .53504 L .55507 .55412 L .57412 .57185 L .59243 .58817 L .61193 .60463 L .63069 .61952 L .65064 .63428 L .66986 .64738 L .68834 .65878 L .69772 .66398 L .70801 .66906 L .71791 .6731 L .72224 .67453 L .72695 .67578 L .72944 .6763 L .73175 .67668 L .73396 .67694 L .73507 .67703 L .73571 .67708 L .7363 .67711 L .73689 .67713 L .73722 .67714 L .73753 .67714 L .73815 .67715 L .7385 .67714 L .73882 .67714 L .73939 .67712 L .74001 .6771 L .74059 .67706 L .74113 .67703 L .74233 .67691 L .74366 .67673 L .74504 .67648 L .74634 .67619 L .74872 .67551 L .75132 .67452 L .75362 .67342 L .75606 .67198 L .76068 .66845 L .76496 .66405 L .7699 .65742 L .77524 .64799 L .78029 .63653 L .78493 .62338 L .79397 .58884 L Mistroke .7991 .5628 L .80389 .53334 L .81367 .45412 L .82258 .35344 L .83174 .21233 L .84171 .00215 L Mfstroke .84171 .00215 m .84179 0 L s .1938 0 m .19788 .05021 L .20257 .0977 L .20752 .13889 L .21638 .19458 L .22093 .21615 L .22581 .23521 L .23461 .26138 L .23949 .27243 L .24481 .28235 L .25446 .2963 L .26396 .30654 L .27404 .31518 L .29216 .32789 L .31146 .34022 L .33003 .35231 L .34977 .36597 L .36879 .38008 L .38709 .39456 L .40656 .41095 L .4253 .42761 L .44522 .44615 L .46441 .46467 L .48287 .4829 L .50251 .50251 L .52143 .52137 L .53961 .53925 L .55897 .55781 L .57761 .57502 L .59742 .59247 L .61651 .60835 L .63487 .62271 L .6544 .637 L .67321 .64984 L .69319 .66274 L .71245 .67515 L .73098 .68891 L .74057 .69801 L .75069 .71058 L .76013 .72681 L .76894 .74808 L .77397 .76401 L .77873 .78239 L .78764 .82818 L .79228 .85963 L .79721 .90029 L .80606 .99736 L s .80606 .99736 m .80625 1 L s .1938 0 m .19788 .05021 L .20257 .0977 L .20752 .13889 L .21638 .19458 L .22093 .21615 L .22581 .23521 L .23461 .26138 L .23949 .27243 L .24481 .28235 L .25446 .2963 L .26396 .30654 L .27404 .31518 L .29216 .32789 L 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"Text", TextAlignment->Right], Cell[TextData[{ "Bekanntlich gilt ", Cell[BoxData[ FormBox[ RowBox[{\(tan(x)\), "=", FractionBox[ StyleBox[\(sin(x)\), FontSize->10], StyleBox[\(cos(x)\), FontSize->10]]}], TraditionalForm]]], " und ", Cell[BoxData[ \(TraditionalForm\`sin(\[Pi]\/4) = cos(\[Pi]\/4)\)]], " , d.h. ", Cell[BoxData[ \(TraditionalForm\`\(\(tan(\[Pi]\/4) = 1\)\(\ \)\(,\)\)\)]], " bzw. ", Cell[BoxData[ FormBox[ RowBox[{\(arctan(1)\), "=", RowBox[{\(\[Pi]\/4\), " ", ".", Cell[""]}]}], TraditionalForm]]], "\n\nAlso gilt ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/4 = \(arctan(1) = \ 1 - 1\/3 + 1\/5 - 1\/7 + \(-\(\(\[CenterEllipsis]\)\(\ \ \)\(.\)\)\)\)\)]], "\n\n(", ButtonBox["James Gregory", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.\ html"], None}, ButtonStyle->"Hyperlink"], ", 1671, Gottfried Wilhelm Leibniz, 1674) \nDiese Reihe konvergiert ", StyleBox["sehr", FontFamily->"Arial", FontSize->10, FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], " langsam, und es ist nicht ratsam damit einen numerischen Wert von \[Pi] \ berechnen zu wollen. Es ist jedoch lange dar\[UDoubleDot]ber nachgedacht \ worden, wie die obige Formel f\[UDoubleDot]r eine numerische Berechnung von \ \[Pi] benutzt werden kann.\nAus dem Additionstheorem f\[UDoubleDot]r den \ Tangens\n\n\t ", Cell[BoxData[ \(TraditionalForm\`tan(x + y)\)]], Cell[BoxData[ FormBox[ RowBox[{"=", FractionBox[ StyleBox[\(tan(x)\ + \ tan(y)\), FontSize->10], StyleBox[\(1\ - \ \ \(tan(x)\)\[CenterDot]\(tan(y)\)\), FontSize->10]]}], TraditionalForm]]], "\n\t\t\nfolgt\t\t ", Cell[BoxData[ \(TraditionalForm\`tan(2 x)\)]], " \t", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FractionBox[ StyleBox[\(\(2\) \( tan(x)\)\(\ \)\), FontSize->10], StyleBox[\(1\ - \ \ \(tan\^2\) (x)\), FontSize->10]], " ", "."}]}], TraditionalForm]]], "\n\nNun sei ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Alpha]\)\(:=\)\(arctan( 1\/5)\)\(\ \ \ \)\)\)]], " \n\nDann folgt\t", Cell[BoxData[ \(TraditionalForm\`tan(2 \[Alpha])\)]], " ", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FractionBox[ StyleBox[\(\(2\) \( tan(\[Alpha])\)\(\ \)\), FontSize->10], StyleBox[\(1\ - \ \ \(tan\^2\) (\[Alpha])\), FontSize->10]], "=", RowBox[{ FractionBox[ StyleBox[\(2\/5\), FontSize->10], StyleBox[\(1\ - \ 1\/25\), FontSize->10]], "=", FractionBox[ StyleBox["5", FontSize->10], StyleBox["12", FontSize->10]]}]}]}], TraditionalForm]]], " \n\nund\t\t", Cell[BoxData[ \(TraditionalForm\`tan(4 \[Alpha])\)]], " ", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FractionBox[ StyleBox[\(\(2\) \( tan(2 \[Alpha])\)\(\ \)\), FontSize->10], StyleBox[ RowBox[{"1", " ", "-", " ", RowBox[{ SuperscriptBox[ StyleBox["tan", FontSize->10], "2"], "(", \(2 \[Alpha]\), ")"}]}], FontSize->10]], "=", RowBox[{ FractionBox[ StyleBox[\(10\/12\), FontSize->10], StyleBox[\(1\ - \ 25\/144\), FontSize->10]], "=", RowBox[{ FractionBox[ StyleBox["120", FontSize->10], StyleBox["119", FontSize->10]], " ", "."}]}]}]}], TraditionalForm]]], "\n\nNun seien ", Cell[BoxData[ FormBox[ RowBox[{\(\[Beta] := arctan(1\/239)\), " ", ",", RowBox[{"x", ":=", " ", RowBox[{\(4 \[Alpha]\ \ und\ y\), ":=", RowBox[{"-", RowBox[{"\[Beta]", " ", ".", Cell[""]}]}]}]}]}], TraditionalForm]]], "\nDann ist ", Cell[BoxData[ \(TraditionalForm\`tan(4 \[Alpha] - \[Beta])\)]], Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FractionBox[ StyleBox[\(tan(x)\ + \ tan(y)\), FontSize->10], StyleBox[\(1\ + \ \ \(tan(x)\)\[CenterDot]\(tan(y)\)\), FontSize->10]], "=", RowBox[{ FractionBox[ StyleBox[\(120\/119\ - \ 1\/239\), FontSize->10], StyleBox[\(1 + 120\/119\[CenterDot]1\/239\), FontSize->10]], "=", RowBox[{ FractionBox[ StyleBox[\(120\[CenterDot]239\ - \ 119\), FontSize->10], StyleBox[\(119\[CenterDot]239\ + \ 120\), FontSize->10]], "=", RowBox[{ FractionBox[ StyleBox[\(119\[CenterDot]239\ \ + \ 239\ - \ 119\), FontSize->10], StyleBox[\(119\[CenterDot]239\ + \ 120\), FontSize->10]], "=", \(\(1\)\(\ \ \)\(.\)\)}]}]}]}]}], TraditionalForm]]], "\n\nAlso gilt ", Cell[BoxData[ FormBox[ RowBox[{\(4 \[Alpha] - \[Beta]\), "=", FractionBox[ StyleBox["\[Pi]", FontSize->10], StyleBox["4", FontSize->10]]}], TraditionalForm]]], " bzw. ", Cell[BoxData[ \(TraditionalForm\`\[Pi] = 16 \[Alpha] - 4 \( \(\[Beta]\)\(\ \ \)\(.\)\)\)]], "\n\nNun ist \t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Alpha]", "=", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox["5", FontSize->10]], "-", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox["3", FontSize->10]], SuperscriptBox[ RowBox[{"(", FractionBox[ StyleBox["1", FontSize->10], StyleBox["5", FontSize->10]], ")"}], "3"]}], "+", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox["5", FontSize->10]], SuperscriptBox[ RowBox[{"(", FractionBox[ StyleBox["1", FontSize->10], StyleBox["5", FontSize->10]], ")"}], "5"]}], "-", \(+\[CenterEllipsis]\)}]}], " ", ",", " ", RowBox[{"\[Beta]", "=", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox["239", FontSize->10]], "-", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], StyleBox["3", FontSize->10]], SuperscriptBox[ RowBox[{"(", FractionBox[ StyleBox["1", FontSize->10], StyleBox["239", FontSize->10]], ")"}], "3"]}], "+", RowBox[{ FractionBox[ StyleBox["1", FontSize->10], "5"], SuperscriptBox[ RowBox[{"(", FractionBox[ StyleBox["1", FontSize->10], StyleBox["239", FontSize->10]], ")"}], "5"]}], "+", \(-\(\(\[CenterEllipsis]\)\(\ \ \)\(.\)\)\)}]}]}], TraditionalForm]]], " \n\nMit diesen Formeln l\[ADoubleDot]\[SZ]t sich \[Pi] gut berechnen. \ Sie wurden von ", ButtonBox["John Machin", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Machin.\ html"], None}, ButtonStyle->"Hyperlink"], " (englischer Astronom, 1680 - 1751) 1706 gefunden.\nIn der folgenden \ Tabelle sind einige Werte eingetragen. ", StyleBox["n ", FontSlant->"Italic"], " in der ersten Spalte gibt dabei jeweils an, wieviele Summanden f\ \[UDoubleDot]r \[Alpha] bzw. \[Beta] berechnet wurden.\nDie ersten 10 \ Dezimalstellen von \[Pi] lauten: 3.141 592 654 .\n\n", Cell[BoxData[ StyleBox[GridBox[{ {\(\(\ \)\(1\)\(\ \)\), \(5359397032/ 1706489875\), \(3.140\ 597\ 029\)}, {\(\(2\)\(\ \)\), \(38279241713339684/ 12184551018734375\), \(3.141\ 621\ 029\)}, {\(\(2\)\(\ \)\), \(76528487109180192540976/ 24359780855939418203125\), \(3.141\ 591\ 772\)}, {\(\(4\)\(\ \)\), \(327853873402258685803048818236/ 104359128170408663038552734375\), \(3.141\ 592\ 682\)}, {\(\(5\)\(\ \)\), \(\(\ \ \)\(\(5150018754384552205994973405622666696/ 1639301884061026141391921953564453125\)\(\ \)\)\), \(\(\ \)\(\ \(3.141\)\(\ \)\(592\)\(\ \)\(653\)\(\ \)\)\)} }, GridFrame->True, RowLines->True, ColumnLines->True], FontFamily->"Times New Roman"]]], "\n\n" }], "Text"] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1400}, {0, 975}}, WindowSize->{556, 656}, WindowMargins->{{0, Automatic}, {Automatic, -2}}, PrintingCopies->1, PrintingPageRange->{9, 9}, PageHeaders->{{Inherited, Inherited, Inherited}, {Cell[ TextData[ "7. 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