Expand the following in a Laurent series valid in the region indicated.
$$e^{1/(z-1)} \; (|z|>1).$$
How can I expand this series? I don't know how to take care of ${1/(z-1)}$ when we need to expand around $0$. The answer says $\sum_{n=1}^\infty \frac{a_n}{z^n}$ where $a_n=\sum_{k=1}^n \binom{n-1}{k-1} \frac{1}{k!}$. I would greatly appreciate any help.