Fourier Series

Obtain Fourier series for \(f(x)=x^3\) in the interval \(-\pi\le x\le\pi\).

Obtain the Fourier Series to represent \(f(x)=x-x^2\) from \(x=-\pi\) to \(x=\pi\) and hence deduce that \(\pi^2/12=1/1^2-1/2^2+1/3^2-1/4^2+\cdots\).

Find the Fourier series for the function \(f(x)=x^2\) in the interval \(-\pi\le x\le\pi\) and deduce that \(\pi^2/12=1/1^2-1/2^2+1/3^2-\cdots\).