You need to look at these in terms of limits and Laurent series and focus on just the term that has the singular component. For example, at z=0, the exp term is a constant so when looking at the function at z=0, you could just as well consider .
If the pole is removable, then exists. So what is ?
If it has a pole of order one, then exists and is not zero. And if it has a pole of order k, then exists and is not zero.
Now, what about all the pi's? They are all the same in terms of singular order right so just consider the one at . What is:
for k=1, k=2, . . . until it exists and is not zero?
The term just goes to so can consider:
That's infinity right?
How about then:
Do a double L'Hospital on it and get one. So it's a pole of order 2.
As far as the , well, has an essential singularity anywhere is singular. You can prove this by writing it's Laurent series.