Find the region of convergence and uniform convergence of the following series $$\sum\limits_{n \geq 0} \frac {z^n} {1 + z^{2n}}.$$
Since $1 + |z|^{2n}\gt |z|^{2n}$ we have $\left | \frac {z^n} {1 + z^{2n}} \right | \lt \left | \frac {z^n} {z^{2n}} \right | = \left | \frac {1} {z^n} \right |.$ Since the geometric series $\sum\limits_{n \geq 0} \frac {1} {z^n}$ converges for $|z| \gt 1$ and diverges for $|z| \lt 1$ it turns out that the region of convergence of the above series is contained in $\{z \in \mathbb C\ |\ |z| \geq 1 \}.$ Now what can we say about the convergence of the given series at $z = a$ with $|a| = 1\ $? First of all $a$ shouldn't be any $m$-th root of unity for $m$ even where there are poles. I am in a fix at this stage. Could anyone give me some suggestions in this regard?
Thanks a bunch.
EDIT $:$ I think that the second inequality what I have written is incorrect. I need to find some different way out.