Triangular wave ($P(s+2\pi )=P(s)$)

A triangular wave is a continuous piecewise linear periodic function whose slope alternates between two values. The even $2\pi$-periodic extension of $bx$ from $[0,\pi ]$ is $P(s)=b|s-2k\pi |$ for $(2k-1)\pi \le s\le (2k+1)\pi$. Its Fourier cosine series contains a constant term and odd modes with coefficients proportional to the inverse square of the mode number. Such extensions describe reflection under Neumann boundary conditions.

The period-two extension of $1-|x|$ on $[-1,1]$ is even. Its Fourier cosine series has $a_0=1$ and $a_n=2{\int }_0^1(1-x)\cos (n\pi x)dx=2[1-(-1{)}^n]/(n^2{\pi }^2)$ by integration by parts. This proves the displayed series, with only odd nonconstant harmonics. Continuity and piecewise smoothness give pointwise equality through the Dirichlet-Jordan convergence theorem; the summable coefficient magnitudes also imply uniform convergence by the Weierstrass M-test. At $x=0$, it gives ${\sum }_{m\ge 0}(2m+1{)}^{-2}={\pi }^2/8$.