... Well, even though I solved that last part, I'm now stuck trying to find the value of . I kind of suspect I should be able to get this from Parseval's Theorem, given in Apostol as . I suspect this because, somehow, the teacher claimed to have used this to show that .
Anyway, I plugged in the constant function and got a truism, so in general I can tell I don't want to use constant functions. I suppose I want to find some function such that is equal to the sum I'm looking to evaluate, but I can't see how to make this magically happen.