In the math.stackexchange.com question How to show this formula using a Fourier sine series expansion? , FMath asked how to prove
using the Fourier sine expansion for the function ; (range: ).
Here is my answer.
To obtain the Fourier sine series expansion for the coefficients must vanish (see below). So let be the odd function extending to the interval defined by
whose graph in is shown in the following figure
We know that the trigonometric Fourier series expansion for in the interval is given by
where the coefficients are the integrals
The integrand is an odd function, while is even. So and
Evaluating this last integral we obtain , which proves that
(the Fourier sine series for the function in the interval ). We see that . We just need to confirm that for this last series reduces to your series. Indeed since
Remark. For a Fourier cosine series we would need to extend to an even function instead, because then would vanish.