I got this question while reading Stein and Shakarchi's Complex Analysis.
Consider the infinite series $$\sum_{n=1}^\infty \frac{2z}{z^2-n^2}.$$ Then it is given in the text that this has simple poles at the integers minus $0$, and no other singularities. However, I cannot prove this fact.
I think this sum converges because for any $z$, we would have some $R>0$ such that $|z|<R$. Then for $n > R$, we have $|\frac{2z}{z^2-n^2}| \le \frac{2R}{n^2-R^2}$. Hence $\sum_{n>N} \frac{2z}{z^2-n^2}$ converges and so does the whole series since the rest is just a finite sum. However I cannot see why this is analytic at all points but the integers except $0$. I would greatly appreciate any help.
Your argument doesn't just prove that your series converges for each $z$ which is not a non-zero integer. It also proves that it converges uniformly on bounded sets, ant therefore that it converges uniformly on compact sets. Therefore, its sum is analytic.