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Math Help - convergent series

  1. #1
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    convergent series

    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.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: convergent series

    Hints: For |z|<1 the series is divergent (use the necessary condition). For |z|>1 the series is convergent (use the ratio test).
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: convergent series

    Quote Originally Posted by Veve View Post
    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
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: convergent series

    Quote Originally Posted by chisigma View Post
    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}
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  5. #5
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    Re: convergent series

    .
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