# Complex Power Series/Examples/n

## Example of Complex Power Series

$S = \ds \sum_{n \mathop \ge 0} n z^n$

has a radius of convergence of $1$.

## Proof

Let $R$ denote the radius of convergence of $S$.

Thus:

 $\ds R$ $=$ $\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {n - 1} n}$ Radius of Convergence from Limit of Sequence $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \cmod {1 - \dfrac 1 n}$ $\ds$ $=$ $\ds 1$ as $\sequence {\dfrac 1 n}$ is a basic null sequence

$\blacksquare$