the theory of convergence series

Question

I have a problem with this question: show that the series $$ \sum_{n=1}^{\infty} \frac{1}{n^2(z-z_n)} $$ when $z_n=e^{ni}$ and $n=1,2,3,\ldots$ is convergent for al values of $z$ which are not on the unit circle $|z|=1$.
My question is why we use geometric representation and how we can solve this question?
Thanks for your helping.

Answer 1

If $z$ is not on the unit circle $C$, then $dist(z,C)=d>0$. Given that $z_n\in C$, we have $\|z-z_n\|\ge d$ and $$ \|\frac{1}{n^2(z-z_n)}\|\le \|\frac{1}{dn^2}\| $$ Since the series $\sum 1/n^2$ is convergent, your series is absolutely convergent.