Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 18 / d / Solution

For a real $2L$-periodic square-integrable function with the convention $f\sim a_0/2+{\sum }_{n\ge 1}(a_n\cos (n\pi x/L)+b_n\sin (n\pi x/L))$, Parseval's identity isIt states that the squared L2 norm equals the sum of the squared components in the orthogonal Fourier basis, with the normalization factors shown. In the present even-function case every $b_n$ vanishes.