convergent series

**Veve** · August 27th 2011, 06:12 AM

I found the following problem:

Determine all z in $\mathbb{C}$ for which the series \sum_{n=0}^\infinity 1/(1-z^n) converges and find the largest domain in which the series converges to an analytic function.

Any hints?

Thanks.

**FernandoRevilla** · August 27th 2011, 06:32 AM

Hints: For $|z|<1$ the series is divergent (use the necessary condition). For $|z|>1$ the series is convergent (use the ratio test).

**chisigma** · August 27th 2011, 07:51 AM

Originally Posted by Veve

I found the following problem:

Determine all z in $\mathbb{C}$ for which the series \sum_{n=0}^\infinity 1/(1-z^n) converges and find the largest domain in which the series converges to an analytic function.

Any hints?

Thanks.

There is only a minor problem: for n=0 is $1-z^{n}=0$ no matter which is z, so that computing $\frac{1}{1-z^{n}}$ is a bit unconfortable...

Kind regards

$\chi$ $\sigma$

**FernandoRevilla** · August 27th 2011, 08:08 AM

Originally Posted by chisigma

so that computing $\frac{1}{1-z^{n}}$ is a bit unconfortable..

Don't waste a problem by a certain and obvious typo in the OP: $\sum_{n=1}^{+\infty}$ instead of $\sum_{n=0}^{+\infty}$

**Duke** · August 27th 2011, 12:57 PM

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