PDE by Fourier solution giving zero coeffiecant

Hi all,

I am working through a PDE and get through to the general solution, but get a seemingly zero Fourier coefficient.

Using seperable variables method and setting the general solution and substituting into original equation:

where is an arbitary constant. Solving the ODE's given yields:

and

->Condition applied

->now apply second condition

since always satisfies the zero condition

rewrite infinite solution as a fourier series.

evaluate

I evaluated the terms to simplify first:

subn

but subn

but subn

now back into integral:

It is this last part I always seem to have the expression collapsing to zero; since and are always the same with the limits of

I know I must have made a mistake somewhere, I suspect in changing the period length and associated limits ,but can't see it!

Many thanks for reading.