Proving series converges using Fourier series
For $0\le x\le\pi$, \begin{align} \sin(x)=|\sin(x)|&=\frac2{\pi}-\frac4{\pi}\sum_{n=1}^\infty\frac{\cos(2nx)}{4n^2-1}\\ &=\frac2{\pi}-\frac2{\pi}\sum_{n=1}^\infty\frac{\cos(2nx)+\cos(-2nx)+i\sin(2nx)+i\sin(-2nx)}{4n^2-1}\\ &=\frac2{\pi}-\frac2{\pi}\left(1+\sum_{n=-\infty}^\infty\frac{\cos(2nx)+i\sin(2nx)}{4n^2-1}\right)\\ &=\frac2{\pi}\sum_{n=-\infty}^\infty\frac{\cos(2nx)+i\sin(2nx)}{1-4n^2}\\ \end{align}