Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 4 / 5A / Solution

For the half-range Fourier cosine series on $(0,1)$, the coefficients areTwo integrations by parts, with $k=n\pi$, giveThusThe even, $2$-periodic extension is continuous and piecewise continuously differentiable, so the Fourier series converges to it, including at $x=1$. Equivalently, the summable coefficient bound gives uniform convergence and permits taking the limit from the interior. Evaluating at $x=1$, where $(-1{)}^n\cos (n\pi )=1$, yieldsThis is the Basel problem.