Evaluating sum using Fourier Series

[tedwillis]

First, I've had to find the Fourier series of $$F(t) = |sin(t)|,$$ which I've calculated as

$$f(t) = \frac{2}{\pi} + \sum_{n=1}^{\infty}\frac{4cos(2nt)}{\pi-4\pi n^2}$$

I'm pretty sure that's right, but now I need to evaluate the sum using the above Fourier series:
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{4n^2-1}$$

I don't really have any clue about where to start.

 

[HallsofIvy]

Well, if
$$f(t)= \frac{2}{\pi}+ \sum_{n= 1}^\infty \frac{4cos(2nt)}{\pi- 4\pi n^2}$$
it follows that
$$-\frac{4}{\pi}\sum_{n=1}^\infty \frac{cos(2nt)}{4n^2- 1}= f(t)- \frac{\pi}{2}$$
so that
$$\sum_{n=1}^\infty \frac{cos(2nt)}{4n^2- 1}= \frac{\pi}{4}(\frac{\pi}{2}- f(t))$$

Now, what should you choose t to be?