Complex analysis - finding the order of poles

I am stuck on the following question:

Find the poles of z/(1-exp(z)) and determine their orders.

I have the poles as being all z*2*Pi*i*k with k an integer other than 0 as I am not sure what happens at 0.

How do I find the order though? It seems to me that writing the series expansion out does not help much.

Thanks for any and all help (my guess is that the order is 1)

further complex pole thinking

I was thinking about this some more, and the function should have a pole at since (also at any multiple of )

The series expansion is correct, but the function still tends to infinity. To show that is a pole we have to be able to show

I can't seem to be able to do that