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Math Help - convergence of a power series

  1. #1
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    convergence of a power series

    I have the following problem: Prove that the power series \sum \frac {z^n}{n} converges at every point of the unit circle except z=1.

    The part with z=1 I already did it, but I have problems with the rest. Is there a way to prove the convergence using Fourier coefficients?

    Thanks.
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  2. #2
    Super Member girdav's Avatar
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    Re: convergence of a power series

    Let s_j:=\sum_{k=0}^jz^k. We have for n\geq 1:
    \begin{align*}\sum_{k=1}^n\frac{z^k}k&=\sum_{k=1}^  n\frac{s_k-s_{k-1}}k\\&=\sum_{j=1}^n\frac{s_j}j-\sum_{j=0}^{n-1}\frac{s_j}{j+1}\\<br />
&=\frac{s_n}n+\sum_{j=1}^ns_j\left(\frac 1j-\frac 1{j+1}\right)\end{align*}
    Now show that for all integer j we have |s_j|\leq \frac 1{\sin\left(\frac{\theta}2\right)}.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: convergence of a power series

    Quote Originally Posted by Veve View Post
    I have the following problem: Prove that the power series \sum \frac {z^n}{n} converges at every point of the unit circle except z=1.

    The part with z=1 I already did it, but I have problems with the rest. Is there a way to prove the convergence using Fourier coefficients?

    Thanks.
    The Abel test for convergence of a series extablishes that...

    a) given a sequence a_{n} such that \forall k the partial sum s_{k}= \sum_{n=1}^{k} a_{n} is bounded...

    b) given a sequence b_{n} so that |b_{n}| is decreasing and \lim_{n \rightarrow \infty} b_{n}=0...

    ... then the series \sum_{n=1}^{\infty} a_{n}\ b_{n} converges. In Your case b_{n}=\frac{1}{n} satisfies to criterion b), so that we have to verify thwe criterion a) for the sequence a_{n}= e^{i n \theta}\, \ \theta \ne 0. The partial sum is...

    s_{k}= \sum_{n=1}^{k} e^{i n \theta} = e^{i \theta}\ \frac{1-e^{i k \theta}}{1-e^{i \theta}} (1)

    ... so that is...

    |s_{k}| = \sqrt{\frac{1-\cos k \theta}{1-\cos \theta}} (2)

    ... and for \theta \ne 0 it is evident that (2) is bounded. Then for Abel's test the series \sum_{n=1}^{\infty} \frac{e^{i \theta}}{n} converges if \theta \ne 0...

    Kind regards

    \chi \sigma
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