# Homework Help: Convergence on the unit circle

1. Sep 30, 2011

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

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

Ʃ $\frac{z^{n}}{n^{2}(1 - z^{n})}$

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
| $\frac{z^{n}}{n^{2}(1 - z^{n})}$ | = $\frac{1}{n^{2}|1 - z^{n}|}$, 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$^{n}$ will land close enough to 1(for infintely many n) so that
$\frac{1}{n^{2}|1 - z^{n}|}$ > 1.

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

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

We know that with |z|=1, z$^{n}$ is dense on the unit circle. Hence for any n there exsists an m that would make
$\frac{1}{(|1 - z^{m}|}$ $\geq$ n$^{2}$ 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