So i've hit my first course in PDEs, and we've been employing Fourier series in our solutions, but never learned about the series in particular. We've just been applying the formulae. I'm a little concerned that I may be off mark in how I applied the Fourier series to evaluate the constant in an infinite sum.

I've got a bit of a long question here that I 'think' i've solved, but I was hoping that one of you fine folks could take a read over my work and let me know if i'm on the right track?

Very much appreciated, thanks!

(PDF is too large, I've uploaded to google docs for viewing. Thanks!)