I need some help finding the Laurent expansion and residue of $$\dfrac{\exp \left(\frac1z \right)}{(1-z)}$$
So far I've done $$\sum_{j=0}^\infty \frac{z^{-j}}{j!} \sum_{k=0}^\infty z^k = \sum_{j=0}^\infty \sum_{k=0}^\infty \frac{z^{k-j}}{j!}$$
but don't know where to go from here. And is it also possible to use Cauchy product when one of the powers is $<0$ and the other is $>0$?