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Math Help - Convergence of a Complex Power Series

  1. #1
    Senior Member roninpro's Avatar
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    Convergence of a Complex Power Series

    Hello everyone. I am asked to show that the power series

    \displaystyle \sum_{n=1}^\infty \frac{z^n}{n}

    converges for |z|\leq 1, except for z=1.

    It is easy to show that the series converges to |z|<1 by the root test. However, it is not clear to me why it should converge on the boundary. I tried putting z=e^{i\theta}, but it does not help.

    I would appreciate any suggestions.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Abel's test - Wikipedia, the free encyclopedia

    Kind regards

    \chi \sigma
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