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Convergence on the unit circle

  1. Sep 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Determine the behavior of convergence on the unit circle, ie |z| = 1 of:

    Ʃ [itex]\frac{z^{n}}{n^{2}(1 - z^{n})}[/itex]

    2. Relevant equations

    Obviously this is divergent then z is a root of unity. The question is what happens when z is not a root of unity.

    3. The attempt at a solution
    My thought was originally that it would converge. Now I think it may diverge. To show this I look at
    | [itex]\frac{z^{n}}{n^{2}(1 - z^{n})}[/itex] | = [itex]\frac{1}{n^{2}|1 - z^{n}|}[/itex], since |z| = 1, and show this does not converge to 0 and hence the series must diverge.

    now for ε = 1. I want to show that z[itex]^{n}[/itex] will land close enough to 1(for infintely many n) so that
    [itex]\frac{1}{n^{2}|1 - z^{n}|}[/itex] > 1.

    or there exists infinitely many n such that [itex]\frac{1}{(|1 - z^{n}|}[/itex] [itex]\geq[/itex] n[itex]^{2}[/itex]

    1. The problem statement, all variables and given/known data

    We know that with |z|=1, z[itex]^{n}[/itex] is dense on the unit circle. Hence for any n there exsists an m that would make
    [itex]\frac{1}{(|1 - z^{m}|}[/itex] [itex]\geq[/itex] n[itex]^{2}[/itex] true.
    Now can we show the stars will allign and get m to equal n infinitely many times.

    Thanks for any help.
     
    Last edited: Sep 30, 2011
  2. jcsd
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