Complex Analysis - Poles/Residues

Hi I was wondering if someone would be able to explain something that was in one of my lectures today:

I have this definition:
f(z) has a pole of order N at $Z_0$ if $\displaystyle f(z)=\sum_{n=-N}^{\infty }a_n(z-z_0)^n,a_{-N}\neq 0$.

The first example was finding the residues of $\displaystyle f(z)=\frac{e^z}{z(1+z^2)}$, and we used this other theorem we have about the residues of poles of order 1.

I don't understand how we know that f(z) has a pole of order 1. I understand that there are poles at z=0,i,-i. If someone could explain this I would be extremely grateful.

without being formal it is best to see by example

order of 1
in the denominator z, z-1, z+i

order of 2
in the denominator z², (z-1)²,( z+i)²

order of 3
in the denominator z³, (z-1)³,( z+i)³

etc

Ah so it's basically look at the powers.
Thanks.