Solution
= Solution
Every sine term vanishes at zero, so \b[$G(0)=0$]. The one-sided values of the periodic extension of $f'$ are $f'_{\rm per}(0-)= -1$ and $f'_{\rm per}(0+)=1$. The <Fourier series> converges at this jump to their midpoint,
$$
\boxed{G(0)=\frac{f'_{\rm per}(0-)+f'_{\rm per}(0+)}2=0.}
$$
This midpoint need not equal the arbitrarily assigned value of the periodic derivative at the endpoint.