Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 4 / a

The function is even, so every sine coefficient is zero. Its mean coefficient isFor $n\ge 1$, compute $a_n=(2/\pi ){\int }_0^{\pi }\sin x\cos (nx)\mathrm{d}x$. The product-to-sum identity makes this integral zero for odd $n$ and gives $-4/(\pi (n^2-1))$ for even $n$. Thus the missing Fourier coefficients are $a_n=0$ for odd $n$ and $b_n=0$ for every $n$, with $a_0=4/\pi$. Equivalently,The Fourier series converges to this continuous periodic function; there is no jump value to average.