Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 1 / 2A (Methods)

The Fourier sine series has coefficientsby integration by parts. ThusIts odd extension is $x$ on $(-L,L)$, continued with period $2L$. The usual Fourier convergence result for a piecewise smooth periodic function gives the value at a continuous point and the mean of one-sided limits at a jump. At $x=3L/2$, subtracting one period gives $-L/2$, a continuous point of this extension. Therefore the sine series converges to $-L/2$ there.

For the Fourier cosine series, the even extension is $|x|$ on $[-L,L]$, continued with period $2L$. Its triangular wave is continuous even at the joining points, so the cosine series converges to that extension everywhere. On the requested interval its graph isThe vertices are $(-2L,0),(-L,L),(0,0),(L,L),(2L,0)$. In particular, its value at $3L/2$ is $L/2$.