Uniform convergence of the given series

Question

In this answer, Using the similar proof, we can see the series uniformly converges on $K\subset (B(0,R)\cap\Bbb C\setminus\Bbb Z)$ with $d(K,\Bbb Z) =\delta>0$, the series $\sum_{n=0}^\infty \frac{z}{n^2-z^2}$ converges. My question is can we drop this $d(K,\Bbb Z)=\delta>0$ condition? I don't know how to choose $z$ properly which shows that we can't drop that condition. Could you help?

Answer 1

If $K \subset (B(0,R)\cap (\mathbb C \setminus Z)$ then $K$ has positive distance from $\{N,N+1,...\}$ for some $N$. So the proof in your link shows that the convergence is uniform.