Analysis

A4

\(Quotientenkriterium[a_, k_] := 
      Module[{lim = 
            Limit[FullSimplify[Abs[\((a /. k -> k + 1)\)\/a]], 
              k -> ∞]}, 
If[
          lim < 1, "konvergent", 
          If[lim > 
              1, "divergent", "Konvergenz nicht entscheidbar"]]
]\)

Quotientenkriterium[((k^2)/(2^k)), k]

Quotientenkriterium[((k^k)/(k!)), k]

Zur Konvergenz bei (a):

(∑_(k = 0)^∞) ((k^2)/(2^k))

Man kann sogar allgemein für |x|<1 zeigen

(∑_(k = 0)^∞) (k^2) (x^k)

A2

(∫_1^∞) 
            (1/(x^α)) \[DifferentialD]x

A3

Arkustangensreihe:

(∑_(k = 0)^∞)((((-1))^k)/(2  k + 1)) 
        (x^(2  k + 1))

Konvergenzradius[a_, k_] := 
      Limit[FullSimplify[Abs[(a/((a /. k -> k + 1)))]], 
        k -> ∞]

Konvergenzradius[((((-1))^k)/(2  k + 1)), k]

Konvergenzradius[((k^k)/(k!)), k]

Lineare Algebra

L4

A = {{1, 2, 3, 4}, {1, 1, -1, -1}, {2, 3, 2, 3}}

RowReduce[A]

B = {{1, 1, 0}, {1, 1, 0}, {0, 0, 1}}

RowReduce[B]

L5

B . A

RowReduce[B . A]

L6

M = {{1, 0, 0, 0}, {0, 1, 0, 2}, {0, 0, 1, 0}, {0, 0, 0, 1}}

A = Array[a, {4, 4}]

M . A

A . M

L7

A = Table[If[j == k, a[k], 0], {k, 10}, {j, 10}]

Inverse[A]