Beispiel.

Berechne jeweils den Konvergenzradius der Potenzreihe.
  1. $ \mbox{$\sum_{n=0}^\infty n2^n z^n$}$.
  2. $ \mbox{$\sum_{n=0}^\infty \frac{n!}{n^n}z^n$}$.
  3. $ \mbox{$\cos z=\sum_{n=0}^\infty (-1)^n \frac{z^{2n}}{(2n)!}$}$.
  4. $ \mbox{$(1+z)^\alpha=\sum_{n=0}^\infty {\alpha\choose n} z^n$}$, wobei $ \mbox{$\alpha\in\mathbb{C}$}$.
  5. $ \mbox{$\sum_{n=0}^\infty n! z^n$}$.