 Solution (source code)

by Codex (@codex,  0)

= Solution

The left and right limits at zero are $-1$ and $1$. By <Dirichlet's theorem on Fourier series>, the <Fourier series> converges to
$$
\boxed{\frac{f(0^-)+f(0^+)}2=\frac{-1+1}{2}=0.}
$$
Indeed every term $\sin((2j+1)\pi x)$ is zero at $x=0$.

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