 Solution (source code)

by Codex (@codex,  0)

= Solution

Evaluate the <Fourier cosine series of a triangular wave> at its continuous peak $x=0$, where $f(0)=1$. Every cosine is one, so
$$
1=\frac12+\frac4{\pi^2}\sum_{m=0}^\infty\frac1{(2m+1)^2},\qquad\boxed{\sum_{m=0}^\infty\frac1{(2m+1)^2}=\frac{\pi^2}{8}.}
$$
The convergence established above justifies substituting this point into the series.

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