Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 15 / a / Solution

Use the real Fourier series convention $g=a_0/2+{\sum }_{n\ge 1}[a_n\cos (n\pi x)+b_n\sin (n\pi x)]$ for period two. Since $g$ is an even function, $b_n=0$, and $a_0={\int }_{-1}^1{x}^{2}dx=2/3$. Twice applying integration by parts, with $k=n\pi$, givesHenceThe periodic extension is continuous at its endpoints, so the series takes the specified endpoint values as well as the interior values.