How to Find B_n in a Fourier Series Problem on the Interval -L < x < L?

[meteorologist1]

Hi, I need help on the following problem on Fourier series:

Let phi(x)=1 for 0<x<pi. Expand
$$1 = \sum\limits_{n = 0}^\infty B_n cos[(n+ \frac{1}{2})x]$$
a) Find B_n.
b) Let -2pi < x < 2pi. For which such x does this series converge? For each such x, what is the sum of the series?
c) Apply Parseval's equality to this series. Use it to calculate the sum
1 + 1/(3^2) + 1/(5^2) + 1/(7^2) + ...

I know the formula for B_n for a function's Fourier series on the interval -L < x < L, so in this question I need to do some kind of odd or even extension for parts a and b, but I don't know how. Please help. Thanks.

 

[meteorologist1]

Ok, actually for part a, I'm pretty sure I should do an even extension of the function at x=0 so that it runs from -pi to pi. And I can then determine the B_n's.

But for part b, it looks like I need to somehow extend to -2pi to 2pi. I looked in books, and does an odd extension at x=pi so that function is -1 from pi to 2i. But I don't understand why. And how would I find the x's for the series to converge?