find the residue of

cos(z)/z^2 at the singularity z=0?

Did you forget the power series for cosine?  The Laurent series you're looking for follows in the obvious way.

I get the larent series

z^-2 - 1 + z^2/4! - z^4/6! + ...

There's no z^-1 term though. Does that mean the residue is 0?

>>3
Yes, that is indeed correct.  Laurent expansion is a valid method of calculating residue for any singularity. (even for an essential singularity)  You can use the formulation involving limits if you are only dealing with poles.