Past exam of the mathematics course of the University of Cambridge / 2013 / ib / Paper 1 / 14B / i

Integration by parts gives the Fourier sine series coefficientsHenceThe odd periodic extension is a sawtooth, equal to $x$ on $(-\pi ,\pi )$ and repeated every $2\pi$; its jumps at odd multiples of $\pi$ have midpoint value zero in the Fourier series. In particular the sine series equals zero, rather than the specified endpoint value $\pi$, at $x=\pi$. The even periodic extension is the continuous triangular wave equal to $|x|$ on $[-\pi ,\pi ]$. Their sketches are below.

The Parseval identity for a Hilbertian basis for this sine expansion gives