Laurent series / residual theorem

[ex3e]

Homework Statement

http://img243.imageshack.us/img243/4339/69855059.jpg

Homework Equations

i've heard that the solution requires the use of the exponential taylor series:
http://img31.imageshack.us/img31/6163/37267605.jpg

The Attempt at a Solution

i know that the first step is to convert cos(1/z) into it's complex number form which is:
cos(1/z) = 0.5 (e^(i/z) + e^(-i/z))
I've tried manipulating 1/z and z^5 into the exp.taylor series but it gets messy (might be doing it wrong). any ideas would be great!

 

[g_edgar]

I would say you don't have to convert the cosine to exponential form if you already know the Maclauren series of cosine.