Fourier cosine series and sum help I have been having some problems with the following problem:
Find the Fourier cosine series of the function $\vert\sin x\vert$ in the interval $(-\pi, \pi)$. Use it to find the sums
$$ \sum_{n\: =\: 1}^{\infty}\:\ \frac{1}{4n^2-1}$$  and $$ \sum_{n\: =\: 1}^{\infty}\:\ \frac{(-1)^n}{4n^2-1}$$
Any help is appreciated, thank you.
edit:I have gotten as far as working out the Fourier cosine series using the equations for cosine series 
$$\phi (X) = 1/2 A_0 +  \sum_{n\: =\: 1}^{\infty}\:\ A_n \cos\left(\frac{n\pi x}{l}\right)$$ and 
$$A_m = \frac{2}{l} \int_{0}^{l} \phi (X) \cos\left(\frac{m\pi x}{l}\right) dx $$
I have found $$A_0 = \frac{4}{l}$$ but the rest of the question is a mess on my end and then I don't know how to relate the rest of it back to those sums.

Hint:
To find the Fourier coefficients of $|\sin(x)|$ note that
$$
\int_{-\pi}^\pi|\sin(x)|\cos(kx)\;\mathrm{d}x=2\int_0^\pi\sin(x)\cos(kx)\;\mathrm{d}x
$$
then use the trig identity $2\sin(x)\cos(y)=\sin(x+y)+\sin(x-y)$.
Once you get the Fourier coefficients you will probably see how to continue.  If not, append your work and say what is causing you difficulty.
