Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 2 / 5D / i

Use the real Fourier series conventionThe constant coefficient is $a_0=2{\int }_0^{1}x(1-x)dx=1/3$. Reflection about $x=1/2$ changes the sine factor's sign and preserves the quadratic, giving $b_n=0$. Two integrations by parts, with $q=2\pi n$, giveConsequently the Fourier series of a periodically extended quadratic arch isThe periodic function is continuous, including at integers where its value is zero. Its piecewise smoothness gives pointwise Fourier series convergence everywhere, and the summable coefficient bound gives uniform and absolute convergence.