How to find the pole order of $z_0=0$ for $\frac{z^3}{1+z-\exp(z)}$

Question

Find the poler order of $z_0=0$ for: $$\frac{z^3}{1+z-\exp(z)}$$

I can't see an easy way with that, is imposible find the Laurent series of that without search $a_n, b_n.$

Answer 1

$e^z=1+z+z^2/2+z^3/6+\dots$. You can factor $z^2$ out of the denominator, and the other factor is $-(1/2+z/6+z^2/24\dots)$. The $z^2$ cancels, and we have $f(0)=0$.