\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
Symbolic Computation of
Formal Power Series in Maxima
(%i1) | batchload ( FPS ) ; |
\[\tag{%o1} C:/Users/bertr/maxima/FPS.mac\]
1 Overview of the method
(%i2) | f : atan ( z ) + cos ( z ) ; |
\[\tag{%o2} \cos{(z)}+\operatorname{atan}(z)\]
(%i3) | DE : HolonomicDE ( f , F ( z ) ) ; |
\[\tag{%o3} \left( {{z}^{2}}+1\right) \, \left( {{z}^{4}}+8 {{z}^{2}}-1\right) \, \left( \frac{{{d}^{4}}}{d {{z}^{4}}} \operatorname{F}(z)\right) +2 z\, \left( {{z}^{4}}+14 {{z}^{2}}-11\right) \, \left( \frac{{{d}^{3}}}{d {{z}^{3}}} \operatorname{F}(z)\right) +\left( {{z}^{2}}+1\right) \, \left( {{z}^{4}}+8 {{z}^{2}}-1\right) \, \left( \frac{{{d}^{2}}}{d {{z}^{2}}} \operatorname{F}(z)\right) +2 z\, \left( {{z}^{4}}+14 {{z}^{2}}-11\right) \, \left( \frac{d}{d z} \operatorname{F}(z)\right) =0\]
(%i4) | RE : DEtoRE ( DE , F ( z ) , a [ n ] ) ; |
\[\tag{%o4} -\left( n+1\right) \, \left( n+2\right) \, \left( n+3\right) \, \left( n+4\right) \, {a_{n+4}}+\left( n+1\right) \, \left( n+2\right) \, \left( 7 {{n}^{2}}-29 n-1\right) \, {a_{n+2}}+n\, \left( 9 {{n}^{3}}-26 {{n}^{2}}+22 n-27\right) \, {a_n}+{a_{n-2}}\, \left( n-2\right) \, \left( {{n}^{3}}-10 {{n}^{2}}+42 n-35\right) +{a_{n-4}}\, \left( n-4\right) \, \left( n-3\right) =0\]
(%i5) | FindRE ( f , z , a [ n ] ) ; |
\[\tag{%o5} -\left( n+1\right) \, \left( n+2\right) \, \left( n+3\right) \, \left( n+4\right) \, {a_{n+4}}+\left( n+1\right) \, \left( n+2\right) \, \left( 7 {{n}^{2}}-29 n-1\right) \, {a_{n+2}}+n\, \left( 9 {{n}^{3}}-26 {{n}^{2}}+22 n-27\right) \, {a_n}+{a_{n-2}}\, \left( n-2\right) \, \left( {{n}^{3}}-10 {{n}^{2}}+42 n-35\right) +{a_{n-4}}\, \left( n-4\right) \, \left( n-3\right) =0\]
(%i6) | mfoldHyper ( RE , a [ n ] ) ; |
\[\tag{%o6} [[2\operatorname{,}\{\frac{{{\left( -1\right) }^{n}}}{n}\operatorname{,}\frac{{{\left( -1\right) }^{n}}}{\left( 2 n\right) \operatorname{!}}\}]]\]
(%i7) | mfoldHyper ( RE , a [ n ] , 2 , 1 ) ; |
\[\tag{%o7} \{\frac{{{\left( -1\right) }^{n}}}{2 n+1}\operatorname{,}\frac{{{\left( -1\right) }^{n}}}{\left( 2 n+1\right) \, \left( 2 n\right) \operatorname{!}}\}\]
(%i8) | FPS ( f , z , n ) ; |
\[\tag{%o8} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{2 n+1}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i9) | FPS ( sin ( z ) / z ^ 8 , z , n ) ; |
\[\tag{%o9} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n-7}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\]
(%i10) | FPS ( cos ( sqrt ( z ) ) + sin ( z ^ ( 1 / 3 ) ) , z , n ) ; |
\[\tag{%o10} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{\frac{2 n+1}{3}}}}{\left( 2 n+1\right) \, \left( 2 n\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i11) | f : log ( 1 + sin ( z ) ) ; |
\[\tag{%o11} \log{\left( \sin{(z)}+1\right) }\]
(%i12) | DE : QDE ( f , F ( z ) ) ; |
\[\tag{%o12} \frac{{{d}^{3}}}{d {{z}^{3}}} \operatorname{F}(z)+\left( \frac{d}{d z} \operatorname{F}(z)\right) \, \left( \frac{{{d}^{2}}}{d {{z}^{2}}} \operatorname{F}(z)\right) =0\]
(%i13) | FindQRE ( f , z , a [ n ] ) ; |
\[\tag{%o13} \left( \sum_{k=0}^{n}{\left. \left( k+1\right) \, \left( k+2\right) \, {a_{k+2}}\, \left( n-k+1\right) \, {a_{n-k+1}}\right.}\right) +\left( n+1\right) \, \left( n+2\right) \, \left( n+3\right) \, {a_{n+3}}=0\]
(%i14) | FPS ( f , z , n ) ; |
\[\tag{%o14} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{n}}\right.}\operatorname{,}{A_{n+4}}=\frac{-\left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, \left( k+2\right) \, {A_{k+2}}\, \left( n-k+2\right) \, {A_{n-k+2}}\right.}\right) -\left( n+2\right) \, \left( n+3\right) \, {A_{n+3}}+\left( n+2\right) \, {A_{n+2}}}{\left( n+2\right) \, \left( n+3\right) \, \left( n+4\right) }\operatorname{,}n\operatorname{> =}0\operatorname{,}[{A_0}=0\operatorname{,}{A_1}=1\operatorname{,}{A_2}=-\frac{1}{2}\operatorname{,}{A_3}=\frac{1}{6}]]\]
(%i15) | FPS ( ( 1 + tan ( z ) ) / ( 1 − tan ( z ) ) , z , n ) ; |
\[\tag{%o15} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{n}}\right.}\operatorname{,}{A_{n+3}}=\frac{2 \left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, {A_{k+1}}\, {A_{n-k+1}}\right.}\right) +2 \left( n+2\right) \, {A_{n+2}}+4 {A_{n+1}}}{\left( n+2\right) \, \left( n+3\right) }\operatorname{,}n\operatorname{> =}0\operatorname{,}[{A_0}=1\operatorname{,}{A_1}=2\operatorname{,}{A_2}=2]]\]
(%i16) | QNF ( tan ( z ) , z , n ) ; |
\[\tag{%o16} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{n}}\right.}\operatorname{,}{A_{n+3}}=\frac{2 \left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, {A_{k+1}}\, {A_{n-k+1}}\right.}\right) +2 {A_{n+1}}}{\left( n+2\right) \, \left( n+3\right) }\operatorname{,}n\operatorname{> =}0\operatorname{,}[{A_0}=0\operatorname{,}{A_1}=1\operatorname{,}{A_2}=0]]\]
(%i17) | FPS ( tan ( z ) , z , n ) ; |
\[\tag{%o17} [\sum_{n=0}^{\infty }{\left. \left( \sum_{k=0}^{n}{\left. \frac{{A_k}\, {{\left( -1\right) }^{n-k}}}{\left( 2 n-2 k+1\right) \operatorname{!}}\right.}\right) \, {{z}^{2 n+1}}\right.}\operatorname{,}{A_k}=\sum_{j=1}^{k}{\left. -\frac{{{\left( -1\right) }^{j}}\, {A_{k-j}}}{\left( 2 j\right) \operatorname{!}}\right.}\operatorname{,}{A_0}=1]\]
(%i18) | FPS ( 1 / ( exp ( z ) − 1 ) , z , n ) ; |
\[\tag{%o18} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{n}}\right.}\operatorname{,}{A_n}=\sum_{k=1}^{n}{\left. -\frac{{A_{n-k}}}{k\operatorname{!}}\right.}\operatorname{,}{A_0}=1]\]
2 Provided commands
3 Classical examples
(%i19) | FPS ( exp ( z ) , z , n ) ; |
\[\tag{%o19} \sum_{n=0}^{\infty }{\left. \frac{{{z}^{n}}}{n\operatorname{!}}\right.}\]
(%i20) | FPS ( log ( 1 + z ) , z , n ) ; |
\[\tag{%o20} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{n+1}}}{n+1}\right.}\]
(%i21) | FPS ( 1 / ( 1 − z ) , z , n ) ; |
\[\tag{%o21} \sum_{n=0}^{\infty }{\left. {{z}^{n}}\right.}\]
(%i22) | FPS ( cos ( z ) , z , n ) ; |
\[\tag{%o22} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i23) | FPS ( atan ( z ) , z , n ) ; |
\[\tag{%o23} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{2 n+1}\right.}\]
(%i24) | FPS ( atanh ( z ) , z , n ) ; |
\[\tag{%o24} \sum_{n=0}^{\infty }{\left. \frac{{{z}^{2 n+1}}}{2 n+1}\right.}\]
(%i25) | FPS ( cosh ( z ) , z , n ) ; |
\[\tag{%o25} \sum_{n=0}^{\infty }{\left. \frac{{{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i26) | FPS ( sinh ( z ) , z , n ) ; |
\[\tag{%o26} \sum_{n=0}^{\infty }{\left. \frac{{{z}^{2 n+1}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\]
(%i27) | FPS ( cos ( z ) , z , n ) ; |
\[\tag{%o27} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i28) | FPS ( sin ( z ) , z , n ) ; |
\[\tag{%o28} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\]
(%i29) | FPS ( sin ( z ) + cos ( z ) , z , n ) ; |
\[\tag{%o29} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i30) | FPS ( asin ( z ) , z , n ) ; |
\[\tag{%o30} \sum_{n=0}^{\infty }{\left. \frac{\left( 2 n\right) \operatorname{!} {{z}^{2 n+1}}}{\left( 2 n+1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}\]
(%i31) | FPS ( asinh ( z ) , z , n ) ; |
\[\tag{%o31} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, \left( 2 n\right) \operatorname{!} {{z}^{2 n+1}}}{\left( 2 n+1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}\]
(%i32) | declare ( p , constant ) $ |
(%i33) | FPS ( 1 / ( 1 − z ) ^ p , z , n ) ; |
\[\tag{%o33} \sum_{n=0}^{\infty }{\left. \frac{{{(p)}_n}\, {{z}^{n}}}{n\operatorname{!}}\right.}\]
4 More thoughtfull series
(%i34) | FPS ( erf ( z ) , z , n ) ; |
\[\tag{%o34} \frac{2 \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) n\operatorname{!}}\right.}}{\sqrt{\ensuremath{\pi} }}\]
(%i35) | FPS ( asin ( z ) ^ 2 / z ^ 2 , z , n ) ; |
\[\tag{%o35} \sum_{n=0}^{\infty }{\left. \frac{{{4}^{n}}\, {{n\operatorname{!}}^{2}}\, {{z}^{2 n}}}{\left( n+1\right) \, \left( 2 n+1\right) \operatorname{!}}\right.}\]
(%i36) | FPS ( exp ( asinh ( z ) ) , z , n ) ; |
\[\tag{%o36} \frac{\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, \left( 2 n\right) \operatorname{!} {{z}^{2 \left( n+1\right) }}}{\left( n+1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}}{2}+z+1\]
(%i37) | FPS ( exp ( asin ( z ) ) , z , n ) ; |
\[\tag{%o37} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -\frac{\% i-1}{2}\right) }_n}\, {{\left( \frac{\% i+1}{2}\right) }_n}\, {{4}^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -\frac{\% i}{2}\right) }_n}\, {{\left( \frac{\% i}{2}\right) }_n}\, {{4}^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i38) | FPS ( sqrt ( ( 1 − sqrt ( 1 − z ) ) / z ) , z , n ) ; |
\[\tag{%o38} \frac{\sum_{n=0}^{\infty }{\left. \frac{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, {{4}^{n}}\, {{z}^{n}}}{\left( 2 n+1\right) \operatorname{!}}\right.}}{\sqrt{2}}\]
(%i39) | FPS ( ( z + sqrt ( 1 + z ^ 2 ) ) ^ p , z , n ) ; |
\[\tag{%o39} p\, \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -\frac{p-1}{2}\right) }_n}\, {{\left( \frac{p+1}{2}\right) }_n}\, {{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -\frac{p}{2}\right) }_n}\, {{\left( \frac{p}{2}\right) }_n}\, {{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i40) | FPS ( exp ( z ) − 2 · exp ( − z / 2 ) · cos ( sqrt ( 3 ) · z / 2 + %pi / 3 ) , z , n ) ; |
\[\tag{%o40} 3 \sum_{n=0}^{\infty }{\left. \frac{{{z}^{3 n+1}}}{{{\left( \frac{2}{3}\right) }_n}\, {{\left( \frac{4}{3}\right) }_n}\, {{27}^{n}} n\operatorname{!}}\right.}\]
(%i41) | FPS ( 1 / 2 · log ( ( 1 + z ) / ( 1 − z ) ) − atan ( z ) , z , n ) ; |
\[\tag{%o41} 2 \sum_{n=0}^{\infty }{\left. \frac{{{z}^{4 n+3}}}{4 n+3}\right.}\]
(%i42) | FPS ( log ( 1 + z + z ^ 2 + z ^ 3 ) , z , n ) ; |
\[\tag{%o42} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 \left( n+1\right) }}}{n+1}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{n+1}}}{n+1}\right.}\]
(%i43) | FPS ( cos ( 4 · acos ( z ) ) , z , n ) ; |
\[\tag{%o43} 8 {{z}^{4}}-8 {{z}^{2}}+1\]
(%i44) | FPS ( asin ( z ) + acos ( z ) , z , n ) ; |
\[\tag{%o44} \frac{\ensuremath{\pi} }{2}\]
(%i45) | FPS ( atan ( ( z + p ) / ( 1 − z · p ) ) , z , n ) ; |
\[\tag{%o45} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{2 n+1}\right.}\right) +\operatorname{atan}(p)\]
(%i46) | FPS ( atan ( ( z + p ) / ( 1 − z · p ) ) − atan ( z ) , z , n ) ; |
\[\tag{%o46} \operatorname{atan}(p)\]
(%i47) | FPS ( sin ( 3 · acos ( z ) ) / sqrt ( 1 − z ^ 2 ) , z , n ) ; |
\[\tag{%o47} 4 {{z}^{2}}-1\]
(%i48) | FPS ( 1 / ( sqrt ( 1 − 4 · z ) ) · ( ( 1 − sqrt ( 1 − 4 · z ) ) / 2 · z ) ^ 2 , z , n ) ; |
\[\tag{%o48} 2 \sum_{n=0}^{\infty }{\left. \frac{\left( 2 n+1\right) \operatorname{!} {{z}^{n+4}}}{\left( n+2\right) \, {{n\operatorname{!}}^{2}}}\right.}\]
(%i49) | FPS ( sqrt ( sqrt ( 8 · z ^ 3 + 1 ) − 1 ) , z , n ) ; |
\[\tag{%o49} 2 \sum_{n=0}^{\infty }{\left. \frac{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, {{\left( -8\right) }^{n}}\, {{4}^{n}}\, {{z}^{\frac{3 \left( 2 n+1\right) }{2}}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\]
(%i50) | FPS ( 1 / ( z + z ^ 2 ) , z , n ) ; |
\[\tag{%o50} \sum_{n=0}^{\infty }{\left. {{\left( -1\right) }^{n}}\, {{z}^{n-1}}\right.}\]
(%i51) | FPS ( cos ( z ) / z ^ 1000 , z , n ) ; |
\[\tag{%o51} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 \left( n-500\right) }}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i52) | FPS ( log ( 1 + z ) + cos ( z ) , z , n ) ; |
\[\tag{%o52} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{n+1}}}{n+1}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i53) | FPS ( cos ( z ) + z · sin ( z ) , z , n ) ; |
\[\tag{%o53} \frac{\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 \left( n+1\right) }}}{\left( n+1\right) \, \left( 2 n\right) \operatorname{!}}\right.}}{2}+1\]
(%i54) | FPS ( sin ( z ) + z · cos ( z ) , z , n ) ; |
\[\tag{%o54} \sum_{n=0}^{\infty }{\left. \frac{2 \left( n+1\right) \, {{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \, \left( 2 n\right) \operatorname{!}}\right.}\]
(%i55) | FPS ( log ( 1 + sqrt ( z ) + z + z ^ ( 3 / 2 ) ) , z , n ) ; |
\[\tag{%o55} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{n+1}}}{n+1}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{\frac{n+1}{2}}}}{n+1}\right.}\]
(%i56) | FPS ( atan ( z ) + exp ( z ^ 2 ) , z , n ) ; |
\[\tag{%o56} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{2 n+1}}}{2 n+1}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{z}^{2 n}}}{n\operatorname{!}}\right.}\]
(%i57) | FPS ( exp ( z ) · cos ( z ) , z , n ) ; |
\[\tag{%o57} \left( \sum_{n=0}^{\infty }{\left. -\frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{4 n+3}}}{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, \left( 2 n+1\right) \, \left( 4 n+1\right) \, \left( 4 n+3\right) \, {{64}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{4 n+1}}}{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, \left( 4 n+1\right) \, {{64}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{4 n}}}{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, {{64}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\]
(%i58) | FPS ( exp ( z ) · sin ( z ) , z , n ) ; |
\[\tag{%o58} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{4 n+3}}}{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, \left( 2 n+1\right) \, \left( 4 n+1\right) \, \left( 4 n+3\right) \, {{64}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{4 n+1}}}{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, \left( 4 n+1\right) \, {{64}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{2 \left( 2 n+1\right) }}}{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, \left( 2 n+1\right) \, \left( 4 n+1\right) \, {{64}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\]
(%i59) | FPS ( ( cos ( z ) + sin ( z ) ) ^ 2 , z , n ) ; |
\[\tag{%o59} 2 \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\right) +1\]
(%i60) | FPS ( ( cos ( z ) + sin ( z ) ) ^ 3 , z , n ) ; |
\[\tag{%o60} \left( \sum_{n=0}^{\infty }{\left. \frac{3 \left( {{\left( -1\right) }^{n}}+{{\left( -9\right) }^{n}}\right) \, {{z}^{2 n+1}}}{2 \left( 2 n+1\right) \, \left( 2 n\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{\left( 3 {{\left( -1\right) }^{n}}-{{\left( -9\right) }^{n}}\right) \, {{z}^{2 n}}}{2 \left( 2 n\right) \operatorname{!}}\right.}\]
(%i61) | FPS ( asech ( z ) , z , n ) ; |
\[\tag{%o61} -\left( \sum_{n=0}^{\infty }{\left. \frac{{{4}^{-n-1}}\, \left( 2 n+1\right) \operatorname{!} {{z}^{2 \left( n+1\right) }}}{{{\left( n+1\right) }^{2}}\, {{n\operatorname{!}}^{2}}}\right.}\right) -\log{(z)}+\log{(2)}\]
(%i62) | FPS ( exp ( z ) + log ( 1 + z ) , z , n , %e ) ; |
\[\tag{%o62} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( \% e+1\right) }^{-n-1}}\, \left( {{\left( -1\right) }^{n}}\, \left( n+1\right) \operatorname{!}+{{\left( \% e+1\right) }^{n}}\, {{\% e}^{\% e+1}} n+{{\left( \% e+1\right) }^{n}}\, {{\% e}^{\% e}} n+{{\left( \% e+1\right) }^{n}}\, {{\% e}^{\% e+1}}+{{\left( \% e+1\right) }^{n}}\, {{\% e}^{\% e}}\right) \, {{\left( z-\% e\right) }^{n+1}}}{\left( n+1\right) \, \left( n+1\right) \operatorname{!}}\right.}\right) +\log{\left( \% e+1\right) }+{{\% e}^{\% e}}\]
(%i63) | FPS ( sin ( z ) + exp ( z ) , z , n , %pi ) ; |
\[\tag{%o63} {{\% e}^{\ensuremath{\pi} }}\, \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( z-\ensuremath{\pi} \right) }^{n}}}{n\operatorname{!}}\right.}\right) -\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{\left( z-\ensuremath{\pi} \right) }^{2 n+1}}}{\left( 2 n+1\right) \, \left( 2 n\right) \operatorname{!}}\right.}\]
(%i64) | FPS ( atan ( z ) , z , n , inf ) ; |
\[\tag{%o64} \frac{\ensuremath{\pi} }{2}-\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{-2 n-1}}}{2 n+1}\right.}\]
(%i65) | FPS ( exp ( 1 / z ) , z , n , inf ) ; |
\[\tag{%o65} \sum_{n=0}^{\infty }{\left. \frac{1}{n\operatorname{!} {{z}^{n}}}\right.}\]
(%i66) | FPS ( z · exp ( − z ) · expintegral_ei ( z ) , z , n , inf ) ; |
\[\tag{%o66} \sum_{n=0}^{\infty }{\left. \frac{n\operatorname{!}}{{{z}^{n}}}\right.}\]
(%i67) | FPS ( sqrt ( %pi ) · exp ( z ) · ( 1 − erf ( sqrt ( z ) ) ) , z , n , inf ) ; |
\[\tag{%o67} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, \left( 2 n\right) \operatorname{!} {{z}^{-n-\frac{1}{2}}}}{{{4}^{n}} n\operatorname{!}}\right.}\]
(%i68) | FPS ( exp ( − 1 / z ) · expintegral_ei ( 1 / z ) / z , z , n ) ; |
\[\tag{%o68} \sum_{n=0}^{\infty }{\left. n\operatorname{!} {{z}^{n}}\right.}\]
(%i69) | FPS ( acos ( z ^ ( 1 / 2 ) ) + exp ( z ^ 2 ) , z , n ) ; |
\[\tag{%o69} \left( \sum_{n=0}^{\infty }{\left. -\frac{\left( 2 n\right) \operatorname{!} {{z}^{\frac{2 n+1}{2}}}}{\left( 2 n+1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. \frac{{{z}^{2 n}}}{n\operatorname{!}}\right.}\right) +\frac{\ensuremath{\pi} }{2}\]
(%i70) | FPS ( exp ( asinh ( z ^ 2 ) ) + 1 / ( 1 − z ^ ( 2 / 3 ) ) , z , n ) ; |
\[\tag{%o70} \left( \sum_{n=0}^{\infty }{\left. -\frac{{{\left( -1\right) }^{n}}\, \left( 2 n\right) \operatorname{!} {{z}^{4 n}}}{\left( 2 n-1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. {{z}^{\frac{2 n}{3}}}\right.}\right) +{{z}^{2}}\]
(%i71) | FPS ( z · log ( z ) ^ 2 + asin ( z ) , z , n ) ; |
\[\tag{%o71} \left( \sum_{n=0}^{\infty }{\left. \frac{{{4}^{-n-1}}\, \left( 2 \left( n+1\right) \right) \operatorname{!} {{z}^{2 n+3}}}{{{\left( n+1\right) }^{2}}\, \left( 2 n+3\right) \, {{n\operatorname{!}}^{2}}}\right.}\right) +z\, \left( {{\log{(z)}}^{2}}+1\right) \]
(%i72) | FPS ( log ( 1 + sqrt ( z ) ) + atan ( z ^ ( 1 / 3 ) ) , z , n ) ; |
\[\tag{%o72} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{\frac{n+1}{2}}}}{n+1}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{\sin{\left( \frac{\ensuremath{\pi} \left( n+1\right) }{2}\right) } {{z}^{\frac{n+1}{3}}}}{n+1}\right.}\]
(%i73) | FPS ( sin ( 2 · z ) + cos ( z ) , z , n , %pi / 2 ) ; |
\[\tag{%o73} -\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, \left( 2 {{4}^{n}}+1\right) \, {{\left( z-\frac{\ensuremath{\pi} }{2}\right) }^{2 n+1}}}{\left( 2 n+1\right) \, \left( 2 n\right) \operatorname{!}}\right.}\]
(%i74) | FPS ( asin ( z ) · cos ( z ) , z , n ) ; |
\[\tag{%o74} \sum_{n=0}^{\infty }{\left. \left( \sum_{k=0}^{n}{\left. \frac{\left( 2 k\right) \operatorname{!} {{\left( -1\right) }^{n-k}}}{\left( 2 k+1\right) \, {{4}^{k}}\, {{k\operatorname{!}}^{2}}\, \left( 2 \left( n-k\right) \right) \operatorname{!}}\right.}\right) \, {{z}^{2 n+1}}\right.}\]
(%i75) | FPS ( atan ( z ) · log ( 1 + z ^ 2 ) , z , n ) ; |
\[\tag{%o75} \sum_{n=0}^{\infty }{\left. {{\left( -1\right) }^{n}}\, \left( \sum_{k=0}^{n}{\left. \frac{1}{\left( 2 k+1\right) \, \left( n-k+1\right) }\right.}\right) \, {{z}^{2 n+3}}\right.}\]
(%i76) | FPS ( log ( 1 + z ) ^ 2 , z , n ) ; |
\[\tag{%o76} \sum_{n=0}^{\infty }{\left. {{\left( -1\right) }^{n}}\, \left( \sum_{k=0}^{n}{\left. \frac{1}{\left( k+1\right) \, \left( n-k+1\right) }\right.}\right) \, {{z}^{n+2}}\right.}\]
(%i77) | FPS ( exp ( z ^ ( 3 / 2 ) ) · asin ( z ^ ( 3 / 4 ) ) , z , n ) ; |
\[\tag{%o77} \sum_{n=0}^{\infty }{\left. \left( \sum_{k=0}^{n}{\left. \frac{{{4}^{k-n}}\, \left( 2 \left( n-k\right) \right) \operatorname{!}}{k\operatorname{!} \left( 2 n-2 k+1\right) \, {{\left( n-k\right) \operatorname{!}}^{2}}}\right.}\right) \, {{z}^{\frac{6 n+3}{4}}}\right.}\]
(%i78) | FPS ( log ( 1 + z ) ^ 5 , z , n ) ; |
\[\tag{%o78} [\left( \sum_{n=1}^{\infty }{\left. \sum_{k=0}^{n-1}{\left. \frac{{A_k}\, \left( 5 n-6 k\right) \, {{\left( -1\right) }^{n-k}}\, {{z}^{n+5}}}{n\, \left( n-k+1\right) }\right.}\right.}\right) +{{z}^{5}}\operatorname{,}{A_0}=1]\]
(%i79) | FPS ( atan ( z ) ^ 2 , z , n ) ; |
\[\tag{%o79} \sum_{n=0}^{\infty }{\left. {{\left( -1\right) }^{n}}\, \left( \sum_{k=0}^{n}{\left. \frac{1}{\left( 2 k+1\right) \, \left( 2 n-2 k+1\right) }\right.}\right) \, {{z}^{2 n+2}}\right.}\]
(%i80) | FPS ( exp ( z + z ^ 2 ) , z , n ) ; |
\[\tag{%o80} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{n}}\right.}\operatorname{,}{A_{n+2}}=\frac{{A_{n+1}}+2 {A_n}}{n+2}\operatorname{,}n\operatorname{> =}0\operatorname{,}[{A_0}=1\operatorname{,}{A_1}=1]]\]
(%i81) | FPS ( exp ( z ^ ( 3 / 2 ) + sqrt ( z ) ) , z , n ) ; |
\[\tag{%o81} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{\frac{n}{2}}}\right.}\operatorname{,}{A_{n+3}}=\frac{{A_{n+2}}+3 {A_n}}{n+3}\operatorname{,}n\operatorname{> =}0\operatorname{,}[{A_0}=1\operatorname{,}{A_1}=1\operatorname{,}{A_2}=\frac{1}{2}]]\]
(%i82) | FPS ( sec ( sqrt ( z ) ) , z , n ) ; |
\[\tag{%o82} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{n}}\right.}\operatorname{,}{A_n}=\sum_{k=1}^{n}{\left. -\frac{{{\left( -1\right) }^{k}}\, {A_{n-k}}}{\left( 2 k\right) \operatorname{!}}\right.}\operatorname{,}{A_0}=1]\]
(%i83) | FPS ( tanh ( z ) , z , n ) ; |
\[\tag{%o83} [\sum_{n=0}^{\infty }{\left. \left( \sum_{k=0}^{n}{\left. \frac{{A_k}}{\left( 2 n-2 k+1\right) \operatorname{!}}\right.}\right) \, {{z}^{2 n+1}}\right.}\operatorname{,}{A_k}=\sum_{j=1}^{k}{\left. -\frac{{A_{k-j}}}{\left( 2 j\right) \operatorname{!}}\right.}\operatorname{,}{A_0}=1]\]
(%i84) | FPS ( exp ( z ) · log ( 1 + z ) , z , n ) ; |
\[\tag{%o84} \sum_{n=0}^{\infty }{\left. \left( \sum_{k=0}^{n}{\left. \frac{{{\left( -1\right) }^{n-k}}}{k\operatorname{!} \left( n-k+1\right) }\right.}\right) \, {{z}^{n+1}}\right.}\]
(%i85) | FPS ( sinh ( log ( 1 + z ) ) , z , n ) ; |
\[\tag{%o85} \left( \sum_{n=0}^{\infty }{\left. -\frac{{{\left( -1\right) }^{n}}\, {{z}^{n}}}{2}\right.}\right) +\frac{z}{2}+\frac{1}{2}\]
(%i86) | FPS ( cosh ( log ( 1 + z ) ) , z , n ) ; |
\[\tag{%o86} \left( \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{z}^{n}}}{2}\right.}\right) +\frac{z}{2}+\frac{1}{2}\]
(%i87) | FPS ( cos ( log ( 1 + z ) ) , z , n ) ; |
\[\tag{%o87} \sum_{n=0}^{\infty }{\left. \frac{\left( \% i {{\left( \% i+1\right) }_n} n-\% i {{\left( 1-\% i\right) }_n} n+{{\left( \% i+1\right) }_n}+{{\left( 1-\% i\right) }_n}\right) \, {{\left( -1\right) }^{n}}\, {{z}^{n}}}{2 \left( {{n}^{2}}+1\right) n\operatorname{!}}\right.}\]
(%i88) | FPS ( sin ( log ( 1 + z ) ) , z , n ) ; |
\[\tag{%o88} \sum_{n=0}^{\infty }{\left. -\frac{\left( {{\left( \% i+1\right) }_{n+1}} n+{{\left( 1-\% i\right) }_{n+1}} n-\% i {{\left( \% i+1\right) }_{n+1}}+{{\left( \% i+1\right) }_{n+1}}+\% i {{\left( 1-\% i\right) }_{n+1}}+{{\left( 1-\% i\right) }_{n+1}}\right) \, {{\left( -1\right) }^{n+1}}\, {{z}^{n+1}}}{2 \left( n+1\right) \, \left( {{n}^{2}}+2 n+2\right) n\operatorname{!}}\right.}\]
(%i89) | FPS ( acos ( z ) ^ 2 , z , n ) ; |
\[\tag{%o89} -\ensuremath{\pi} \left( \sum_{n=0}^{\infty }{\left. \frac{\left( 2 n\right) \operatorname{!} {{z}^{2 n+1}}}{\left( 2 n+1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. \frac{{{4}^{n}}\, {{n\operatorname{!}}^{2}}\, {{z}^{2 \left( n+1\right) }}}{\left( n+1\right) \, \left( 2 n+1\right) \operatorname{!}}\right.}\right) +\frac{{{\ensuremath{\pi} }^{2}}}{4}\]
(%i90) | FPS ( asin ( z ) ^ 2 , z , n ) ; |
\[\tag{%o90} \sum_{n=0}^{\infty }{\left. \frac{{{4}^{n}}\, {{n\operatorname{!}}^{2}}\, {{z}^{2 \left( n+1\right) }}}{\left( n+1\right) \, \left( 2 n+1\right) \operatorname{!}}\right.}\]
(%i91) | FPS ( acos ( z ) ^ 2 − asin ( z ) ^ 2 , z , n ) ; |
\[\tag{%o91} \frac{{{\ensuremath{\pi} }^{2}}}{4}-\ensuremath{\pi} \sum_{n=0}^{\infty }{\left. \frac{\left( 2 n\right) \operatorname{!} {{z}^{2 n+1}}}{\left( 2 n+1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}\]
(%i92) | FPS ( sin ( z ) + sinh ( z ) , z , n ) ; |
\[\tag{%o92} 2 \sum_{n=0}^{\infty }{\left. \frac{{{4}^{n}}\, {{z}^{4 n+1}}}{{{\left( \frac{3}{4}\right) }_n}\, {{\left( \frac{5}{4}\right) }_n}\, {{256}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\]
(%i93) | FPS ( cos ( 2 · z ) + sin ( 3 · z ) , z , n ) ; |
\[\tag{%o93} \left( \sum_{n=0}^{\infty }{\left. \frac{3 {{\left( -9\right) }^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \, \left( 2 n\right) \operatorname{!}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{2 n}}}{\left( 2 n\right) \operatorname{!}}\right.}\]
(%i94) | FPS ( sin ( z ) · cos ( z ) , z , n ) ; |
\[\tag{%o94} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{2 n+1}}}{\left( 2 n+1\right) \operatorname{!}}\right.}\]
(%i95) | FPS ( sinh ( z ) · sin ( z ) , z , n ) ; |
\[\tag{%o95} 2 \sum_{n=0}^{\infty }{\left. \frac{\left( n+1\right) \, {{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{2 \left( 2 n+1\right) }}}{{{\left( \frac{3}{4}\right) }_n}\, {{\left( \frac{5}{4}\right) }_n}\, {{64}^{n}}\, \left( 2 \left( n+1\right) \right) \operatorname{!}}\right.}\]
(%i96) | FPS ( cosh ( z ) · cos ( z ) , z , n ) ; |
\[\tag{%o96} \sum_{n=0}^{\infty }{\left. \frac{{{\left( -1\right) }^{n}}\, {{4}^{n}}\, {{z}^{4 n}}}{{{\left( \frac{1}{4}\right) }_n}\, {{\left( \frac{3}{4}\right) }_n}\, {{64}^{n}}\, \left( 2 n\right) \operatorname{!}}\right.}\]
(%i97) | FPS ( exp ( z ) · cosh ( z ) , z , n ) ; |
\[\tag{%o97} \left( \sum_{n=0}^{\infty }{\left. \frac{{{2}^{n-1}}\, {{z}^{n}}}{n\operatorname{!}}\right.}\right) +1\]
(%i98) | FPS ( atanh ( z ) + sqrt ( 1 + z ) + 1 / sqrt ( 1 + z ) , z , n ) ; |
\[\tag{%o98} \left( \sum_{n=0}^{\infty }{\left. \frac{{{z}^{2 n+1}}}{2 n+1}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{2 \left( n-1\right) \, {{\left( -1\right) }^{n}}\, \left( 2 n\right) \operatorname{!} {{z}^{n}}}{\left( 2 n-1\right) \, {{4}^{n}}\, {{n\operatorname{!}}^{2}}}\right.}\]
(%i99) | FPS ( sinh ( z ) ^ 2 + cosh ( z ) ^ 2 , z , n ) ; |
\[\tag{%o99} 2 \left( \sum_{n=0}^{\infty }{\left. \frac{{{4}^{n}}\, {{z}^{2 \left( n+1\right) }}}{\left( n+1\right) \, \left( 2 n+1\right) \operatorname{!}}\right.}\right) +1\]
(%i100) | FPS ( − cosh ( z ) ^ 2 + sinh ( z ) ^ 2 , z , n ) ; |
\[\tag{%o100} -1\]
(%i101) | declare ( q , constant ) $ |
(%i102) | FPS ( 1 / ( p − z ^ 2 ) / ( q − z ^ 3 ) , z , n ) ; |
\[\tag{%o102} \left( \sum_{n=0}^{\infty }{\left. -\frac{p\, {{q}^{-n-1}}\, {{z}^{3 n+2}}}{{{q}^{2}}-{{p}^{3}}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. -\frac{{{z}^{3 n+1}}}{\left( {{q}^{2}}-{{p}^{3}}\right) \, {{q}^{n}}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. \frac{{{z}^{2 n+1}}}{\left( {{q}^{2}}-{{p}^{3}}\right) \, {{p}^{n}}}\right.}\right) +\left( \sum_{n=0}^{\infty }{\left. -\frac{{{p}^{2}}\, {{q}^{-n-1}}\, {{z}^{3 n}}}{{{q}^{2}}-{{p}^{3}}}\right.}\right) +\sum_{n=0}^{\infty }{\left. \frac{q\, {{p}^{-n-1}}\, {{z}^{2 n}}}{{{q}^{2}}-{{p}^{3}}}\right.}\]
(%i103) | FPS ( exp ( 2 · atanh ( sin ( 2 · z ) / ( 1 + cos ( 2 · z ) ) ) ) , z , n ) ; |
\[\tag{%o103} [\sum_{n=0}^{\infty }{\left. {A_n}\, {{z}^{n}}\right.}\operatorname{,}{A_{n+4}}=\frac{3 \left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, \left( k+2\right) \, {A_{k+2}}\, \left( n-k+2\right) \, {A_{n-k+2}}\right.}\right) -\left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, \left( k+2\right) \, \left( k+3\right) \, {A_{k+3}}\, {A_{n-k+1}}\right.}\right) -4 \left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, {A_{k+1}}\, {A_{n-k+1}}\right.}\right) +6 \left( n+2\right) \, \left( n+3\right) \, {A_{n+3}}+8 \left( n+2\right) \, {A_{n+2}}-24 {A_{n+1}}}{\left( n+2\right) \, \left( n+3\right) \, \left( n+4\right) }\operatorname{(}3 \left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, \left( k+2\right) \, {A_{k+2}}\, \left( n-k+2\right) \, {A_{n-k+2}}\right.}\right) -\left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, \left( k+2\right) \, \left( k+3\right) \, {A_{k+3}}\, {A_{n-k+1}}\right.}\right) -4 \left( \sum_{k=1}^{n}{\left. \left( k+1\right) \, {A_{k+1}}\, {A_{n-k+1}}\right.}\right) +6 \left( n+2\right) \, \left( n+3\right) \, {A_{n+3}}+8 \left( n+2\right) \, {A_{n+2}}-24 {A_{n+1}}\operatorname{)}/\left( \left( n+2\right) \, \left( n+3\right) \, \left( n+4\right) \right) \operatorname{,}n\operatorname{> =}0\operatorname{,}[{A_0}=1\operatorname{,}{A_1}=2\operatorname{,}{A_2}=2\operatorname{,}{A_3}=\frac{8}{3}]]\]
(%i104) | FPS ( ( 1 + tan ( z ) ) / ( 1 − tan ( z ) ) − exp ( 2 · atanh ( sin ( 2 · z ) / ( 1 + cos ( 2 · z ) ) ) ) , z , n ) ; |
\[\tag{%o104} 0\]
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Bertrand Teguia Tabuguia