I would like to show $$\sum_{|n|>N}(\frac{1}{z-n}+\frac{1}{n})$$ is holomorphic on $|z|<n$.
To show this, I try to use the same method for Weierstrass $P-$function, so I take the sum in the bracket, tells me that $|\frac{z}{(z-n)n}|$. But I cannot give a good approximation to proceed.
If we prove that that partial sums converge uniformly on compact subsets of $(|z| <N)$ it would follow that the infinite sum is holomorphic. If $K$ is a compact set in this domain then $|\frac 1 {z-n} +\frac 1 n|\leq \frac {|z|} {n(n-|z|)} \leq \frac {C} {n(n-C)}$ for some finite constant $C$. By comparison with $\sum \frac {C} {n(n-C)}$ we see that the series converges uniformly on $K$.