Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 2 / 3D (Methods)

Square the Fourier series, integrate, and use orthogonality; cross terms vanish. This gives ${\int }_{-1}^1|f{|}^2=a_0^2/2+{\sum }_{n\ge 1}(a_n^2+b_n^2)$.

For $f=x^2$, ${\int }_{-1}^1{x}^{4}dx=2/5$, $a_0=2/3$, and $a_n=4(-1{)}^n/({\pi }^2{n}^2)$. Parseval yields $2/5=2/9+16{\pi }^{-4}\sum n^{-4}$, hence $\sum n^{-4}={\pi }^4/90$.