I want to construct a meromorphic function in $\mathbb{C}$ with poles in $z=-n$ for $n=1,2,...$ and residue $n$.
I know that $\sum\limits_{n=1}^{\infty}\frac{n}{z+n}$ satisfies those conditions, but the series doesn't converge uniformly in the compacts of $\mathbb{C}\backslash\mathbb{Z_-}$. I know I have to add some polynomials $p_n(z)$ so that $\sum\limits_{n=1}^{\infty}\left[\frac{n}{z+n}-p_n(z)\right]$ converges uniformly.
What are the simplest polynomials $p_n$ I can choose? Thanks.