Proof that sum^infty of 1/n^2 = ...

I am trying to prove . I've done quite a lot:

Taking derivative of both sides and using part (a), we see that and differ by at most a constant on (in part (a) I have basically shown that the derivative of each side is equal). Evaluating at , we see that the constant difference is , which proves the equality. Likewise evaluating at 0 and we obtain the identity.

(Note, this argument depends on the claim that , at the point when I evaluate at . ...

So here's where I'm stuck. I have a hint that I'm supposed to use the following theorem:

If a function is continuous and periodic on with period , then, using the usual notation about and , .

My first question is, since my function isn't periodic, how do I use this theorem?

Anyway, I started trying this for the function I'm working with, but what I get is:

Now I have to say, already this looks messed up. Am I doing something wrong?