Let $(a_n(z))$ is a sequence of holomorphic functions defined on $\mathbb{C}\setminus A$, where $A$ is a set of simple poles. I am thinking about proving that $\sum_{n=1}^{\infty}\left | a_n(z) \right |$ is uniformly convergent on every compact subset of $\mathbb{C}\setminus A$.
If you have proven that $\left | a_n(z) \right |\leq M_n$, independent of $z$, for all $n\geq N$ (for suitable $N$) on a disc, and that $\sum_{n=N}^{\infty}M_n<\infty$, would that answer the question about the uniform convergence? What about $\sum_{n=1}^{N-1}\left | a_n(z) \right |$?