[SOLVED] Poles, order of it and residue

I must prove that the singularities of are poles. I also must determine their order and the corresponding residue.

.

My attempt :

I'm stuck on the first part. (So I'd appreciate help only on this part, for the moment).

So I wrote .

So if I can find the order of the zeros of I'm done.

I've found out that

The first term of the series really bothers me. If it wasn't there then would be equal to with and so would be a of order of , thus a pole of order of . But of course I should take in count the term and look for , , but I'm stuck.