# Thread: convergent series

1. ## convergent series

I found the following problem:

Determine all z in $\displaystyle \mathbb{C}$ 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.

2. ## Re: convergent series

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

3. ## Re: convergent series

Originally Posted by Veve
I found the following problem:

Determine all z in $\displaystyle \mathbb{C}$ 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 $\displaystyle 1-z^{n}=0$ no matter which is z, so that computing $\displaystyle \frac{1}{1-z^{n}}$ is a bit unconfortable...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

4. ## Re: convergent series

Originally Posted by chisigma
so that computing $\displaystyle \frac{1}{1-z^{n}}$ is a bit unconfortable..
Don't waste a problem by a certain and obvious typo in the OP: $\displaystyle \sum_{n=1}^{+\infty}$ instead of $\displaystyle \sum_{n=0}^{+\infty}$

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