Uniformly convergence of series on compact set

Question

Prove that the series summation $$ \sum_{n=-\infty}^\infty \frac{1}{(z-n)^2} $$ converges uniformly on compact sets.

I am struggling on this problem in complex analysis. I just know uniformly convergence of real series but totally unfamiliar with complex series. How to get start ?

Answer 1

Hint: Weierstrass $M$-test works also for complex series. Fix a compact set $K$. If $z \in K$ and $|n|$ is large enough, the denominator doesn't vanish on $K$. Estimate the supremum norm of $$ \frac{1}{(z-n)^2} $$ for large $n$ on $K$. (Note that the sum isn't defined for $z \in \mathbb{Z}$.)