Convergence on the unit circle

[Trevor Vadas]

Homework Statement

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

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

Homework 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.

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 infinitely 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]

Homework Statement

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.

 

[mighty2000]

Thank you for your question. When considering the behavior of convergence on the unit circle, we must first determine the values of z for which the series will converge. As you correctly noted, the series will diverge when z is a root of unity. This is because when z is a root of unity, z^n will cycle through the same values and the denominator, 1-z^n, will approach 0, making the series divergent.

However, when z is not a root of unity, the behavior of the series becomes more complicated. We can use the ratio test to determine the behavior of convergence for this series. The ratio test states that if the limit of |a_{n+1}/a_n| as n approaches infinity is less than 1, then the series will converge. If the limit is greater than 1, then the series will diverge. If the limit is equal to 1, the test is inconclusive and we must use other methods to determine the behavior of convergence.

Applying this test to our series, we have:

| \frac{z^{n+1}}{(n+1)^{2}(1 - z^{n+1})} | / | \frac{z^{n}}{n^{2}(1 - z^{n})} | = | \frac{z}{(n+1)^{2}}| * | \frac{1 - z^{n}}{1 - z^{n+1}}|

Since |z| = 1, we can simplify this to:

| \frac{1 - z^{n}}{1 - z^{n+1}}| = | \frac{1 - z^{n}}{1 - z^{n}z}| = 1

This means that the limit of |a_{n+1}/a_n| is equal to 1, and the ratio test is inconclusive. Therefore, we must use other methods to determine the behavior of convergence.

One method we can use is the root test, which states that if the limit of the nth root of |a_n| as n approaches infinity is less than 1, then the series will converge. If the limit is greater than 1, then the series will diverge. If the limit is equal to 1, the test is inconclusive.

Applying this test to our series, we have:

lim_{n->∞} (|