Laurent Series of $~\dfrac{ze^z}{z-1}~$

Question

I've come across a problem that, the question is as follows.

Determine the Laurent Series of $~\dfrac{ze^z}{z-1}~$ about $~z=1~$.

I've tried to this in numerous different ways, but I can't seem to make much progress on it.

How would I get started on doing this problem? Any help is much appreciated.

Answer 1

Hint: $\frac{ze^z}{z-1}=e \cdot e^{z-1}(1+ \frac{1}{z-1}).$ Now invoke the power series of $e^{z-1}.$