Find a meromorphic function with poles at $z_n=n+\sqrt{n}$ with $n\in\{1,2,3,\ldots\}$ such that $$f(z)-\frac{1}{(z-z_n)^2}+\frac{1}{z-z_n}$$ has a removable singularity at $z_n$.
I believe that I have to prove that $$f(z)=\sum_{n=1}^\infty \frac{1}{(z-z_n)^2}-\frac{1}{z-z_n}$$ is a meromorphic function; i.e, the series converges uniformly in compacts sets, but I don't know how. Any suggestions?