I am not able to solve this further, this is my approach to the question. I know that the series will look like in the form of powers of "$z$" and I am also about Laurent expansion of $1/(1-z)$ for $|z| < 1$.
But here the " $i$ " inside $1/(z+i)$ and so on, is giving me some trouble, can I take it outside the bracket and then apply Laurent expansion for $1/(1-z)$? It will still hold?