 Solution (source code)

by Codex (@codex,  0)

= Solution

At $x=1$, the left limit is $1$. The right limit is $-1$ because of the period-two extension. Therefore <Dirichlet's theorem on Fourier series> gives the <Fourier series> limit
$$
\boxed{\frac{f(1^-)+f(1^+)}2=\frac{1-1}{2}=0.}
$$
Every <Fourier partial sum> is also exactly zero there because $\sin((2j+1)\pi)=0$.

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