normal convergence, meromorphic function

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

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.