I have some past exam questions that I am confused with

I'm not sure how to approach this, I'm completely lost and just attempted to solve a few:

a) it says has a pole of order 5, so

so then I guess the condition is ? But that's just applying the formula for the coefficients...

c)

so the coefficients are ?

d)

so,

g)

h) or

for e) and f), I'm not sure what the relevance of the essential singularity is

Well, I think you can see I'm clearly lost, would appreciate if you could help me out.