This a question from Fourier Series:
Show that for $0<x<\pi$
First of all, the interval given is an open interval i.e. $(0,\pi)$, but I have read that Fourier Series are only applicable for closed intervals. Then how can I solve this question using Fourier Series ?
Finding out the Fourier coefficients considering the interval $[-\pi,\pi]$, I have calculated the following:
But I think all this not useful in this question as the interval given is different. Also, even if I consider the Fourier series in the interval $[0,\pi]$, then also I will not be able to write the equality ($=$) sign in the equality to be shown in the question because equality means the series converges to the function $x(\pi-x)$ and for convergence of the Fourier series, the initial assumption is that the given function is a periodic function of periodicity $2\pi$. But here the function $x(\pi-x)$ is defined over a period of $\pi$.
Can anyone help me out here ? I will be highly grateful.,