Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 11 / b / Solution

Because the function is even, all sine Fourier coefficients vanish. Its average is zero, so $a_0=0$. The Fourier cosine coefficients of a sine mode follow from the product-to-sum identity:For $n=2$, the integral is directly zero rather than an undefined substitution into a denominator. ThusThe Fourier series equals the continuous function at every point, including the corners. Its coefficients decay quadratically, so the series is also absolutely convergent.