Region of convergence of complex series $\sum \frac{z^n}{1-z^n}$.

Question

I am looking for region of convergence of the complex series $\sum\frac{z^n}{1-z^n}.$ For $|z|>1$ it’s not convergent as $nth$ doesn’t tends to zero . For me it’s seems that it doesn’t convergent for $|z|=1$. Please suggest me for the region $|z|\leq 1$. Thanks

Answer 1

If $|z|=1$ and $z^{n} \neq 1$ for any $n$ (so that the series is defined) then $|\frac {z^{n}} {1-z^{n}}|=\frac 1 {|1-z^{n}|}$ which does not tend to $0$, so the series is not convergent. If $|z|<1$ then $|\frac {z^{n}} {1-z^{n}}|\leq \frac {|z|^{n}} {1-|z|^{n}}$ and $1-|z|^{n} \geq \frac 1 2$ for $n$ sufficiently large. By comparison with $\sum z^{n}$ the series is convergent for $|z| <1$.