Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 2 / 5B / Solution

The Fourier sine series coefficients of $x$ on $(0,\pi )$ areby integration by parts. The odd periodic extension is piecewise smooth, so the Fourier series converges to $x$ at every interior point. HenceTo justify integration from zero, the Fourier sine series partial sums converge to $x$ in L2 norm. By the Cauchy-Schwarz inequality, integration from zero to any $x\in [0,\pi ]$ has error at most $\sqrt{\pi }$ times that L2 norm error, uniformly in $x$. Integrating the partial sums and taking the limit therefore givesThe integrated series is absolutely and uniformly convergent, since its terms are bounded by $4/n^2$. Thus the resulting Fourier cosine series hasIndependently, the constant coefficient isComparison yields the requested exact sum