Question about Complex Analysis and Residue.

Question

I am studying Complex Analysis and I need to compute the residue of $f(z) = exp (1 + \frac{1}{z})$ on $z_0=0$

I Tried for Taylor Series,that is, $exp (1 + \frac{1}{z}) = \displaystyle \sum_{n=0}^{\infty} \frac{1}{n!} \left(1+\frac{1}{z}\right)^n$ and I need to rewrite that sum like Laurent Series , so the residue is the constant ''a'' that appears multiplying $\frac{1}{z-0}$, but I can't separate the expressions and I don't believe that the binomial theorem is the way.

Then I think that I will use the formula $Res[f,0] = \frac{1}{(k-1)!} \displaystyle \lim_{z \to 0} \frac{d^{k-1}}{dz^{k-1}}z^kf(z)$, where $k$ is the order of pole $z=0$, but I need to find the order of $z_0=0$ and this is impossíble...

Then, How to solve it?

Answer 1

Note $e^\left(1+\frac{1}{z}\right)=e . e^{\frac{1}{z}}$. Now do the Laurent series for $e^{1/z}$ as $\sum_{n=0}\frac{1}{n!}\left(\frac{1}{z}\right)^n$. Hopefully now you will get the residue.