normal convergence, meromorphic function

Let $\{ z_k \}$ be a sequence of distinct points such that $|z_k| \rightarrow \infty$ and $\sum_{k=1}^{\infty} |z_k|^{-m-1}< \infty$ . Show that $z^m \sum_{k=1}^{\infty} \frac{1}{z_k^m(z-z_k)}$ converges normally to a meromorphic function with principal part $\frac{1}{z-z_k}$ at $z_k$ . (If $z_k=0$ , we replace the corresponding summand by $\frac{1}{z}$ .)

In this section, we covered the Mittag-Leffler Theorem. I am not sure if we need to apply that result here though. I am not sure what this sequence would converge to. I would appreciate some help with this. Thank you.