Properties of series $\sum_{n=1}^\infty\frac{2z}{z^2-n^2}$
Solution 1:
Fix $k$. For $|z-k| <\frac 1 2$ the series $\sum_{n \neq k} |\frac {2z}{z^{2}-n^{2}}|$ is dominated by $\sum_{n \neq k} \frac {2|k|+1} {n^{2}-(|k|+\frac 1 2 )^{2}}$ and hence this sum is analytic in $|z-k| <\frac 1 2$.