Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 4 / b / Solution

For period two, write $f\sim a_0/2+{\sum }_{n\ge 1}[a_n\cos (n\pi x)+b_n\sin (n\pi x)]$. Direct integration gives $a_0=1/2$. For $0<x<1$, $f(x)+f(-x)=2-3x$ and $f(x)-f(-x)=x$, soThus the Fourier series isIt converges to the values specified in part a. Changing the assigned values at the jumps would not change these integral coefficients.