1. ## laurent series

I need a small bit of help finding the laurent series of

(e^Z)/(Z-Z^2) that converge for 0 < |Z| < R. I also need to find R.

i took z out of the bottom line leaving me with (1/z)(e^z/1-z). As far as i know i should only be interested in the case of z=1 as it's the first non-analytic point after z = 0. Correct?.

Im kinda lost on how to continue this problem.. should i be using the maclaurin series somewhere?

thanks

2. Use the geometric series to argue that $\displaystyle \frac{1}{1- z}= \sum_{n=0}^\infty z^n$. For what values of z does that converge? Dividing each term by z gives $\displaystyle \frac{1}{z(1- z}= \sum_{n=-1}^\infty$, a "Laurent" series since it now includes a negative power.

Of course $\displaystyle e^z= \sum_{n=0}^\infty \frac{z^n}{n!}$. Multiply those two series term by term.

Alternatively find the MacLaurin series for the function $\displaystyle \frac{e^z}{1- z}$ which is analytic at z= 0. Then divide each term by z to get the Laurent series for $\displaystyle \frac{e^z}{z(1- z)}$