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

Every sine term vanishes at zero, so $G(0)=0$. The one-sided values of the periodic extension of $f^{\prime }$ are $f_{\mathrm{per}}^{\prime }(0-)=-1$ and $f_{\mathrm{per}}^{\prime }(0+)=1$. The Fourier series converges at this jump to their midpoint,This midpoint need not equal the arbitrarily assigned value of the periodic derivative at the endpoint.