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for the fourier series of e^x, determine its convergence, and use it to find coth(Pi)

The question had part (a) - determine the complex fourier series of e^x on [-Pi, Pi]

My solution to this one was horrible -

Attachment 24623

for the second part, I determined that the series converged pointwise, but the lecture notes said 'for all x' - I assume this means all x in R, since the notes didn't specify?

However, for the uniform convergence, the only information I could find related to a function that was periodic, and e^x is not periodic, so I'm not sure what to do for that.

For the third part, the problem is to show coth(pi)=1/Pi (Sum(1/(1+n^2))) for n ranging from 0 to infinity...

I know that I have to use coth(pi)=(e^(pi)+e^(-pi))/(e^(pi)-e(^-pi))

But I'm fairly sure that without simplifying my expression for e^x, I wouldn't be able to solve this...

I guess I have to use complex conjugates to normalise the denominators in my Fourier series, but other than that...