So i have a question which asks to find the fourier series of $\left\vert\,\sin\left(x\right)\,\right\vert\,$. I have worked out the solution as $$ {2 \over\pi} - {4 \over \pi}\sum_{k = 1}^{\infty}{\cos\left(2kx\right) \over 4k^{2} - 1} $$
Which i am pretty sure is correct as i have the solution in my book.
The second part of the question asks to work out the sums of $$ \sum_{k = 1}^{\infty}{1 \over 4k^{2} - 1}\qquad\mbox{and}\qquad \sum_{k = 1}^{\infty}{\left(-1\right)^{k} \over 4k^{2} - 1} $$
Im sure this is probably very simple but i have no solution for this and I am struggling to search for an explanation of how to do this on google. Could someone please tell me know it is done ?.
Many thanks.