Region of convergence of a series of finite terms

Question

Say I want to expand \begin{equation} \frac{z^3}{(z+i)^2} \end{equation} about $z=-i$. The answer is straightforward, I get a few terms \begin{equation} \frac{i}{(z+i)^2} - \frac3{z+i}- 3i +(z+i). \end{equation} I am being asked about the region of convergence of these terms. I have thought of two different solutions:

1. There is no region of convergence, this is only relevant for a power series and this isn't one.

2. The series (as much as one can call it) converges to itself.

What's the best way to approach this?

Answer 1

You have a Laurent series. It's just the case that most of the terms vanish. The "series" converges for all $z \neq -i$.