Clarification of what should I do in this Laurent series expansion

Question

For $z \in \mathbb{C}$, I am asked to expand $$ \frac{1}{z-2}-\frac{1}{z-3} $$ into Laurent series around $z_0 =2$. I know how to handle the right term, but how about the left one? should I expand it too? and if so how? It is not clear to me what the final answer should look like.

Answer 1

You don't expand it, since it is already of the form $\sum_{n=-\infty}^\infty a_n(z-2)^n$; you simply take$$a_n=\begin{cases}1&\text{ if }n=-1\\0&\text{ otherwise.}\end{cases}$$