= Fourier cosine series of a triangular wave
{title2=$1/2+(4/\pi^2)\sum_{m\geq0}\cos((2m+1)\pi x)/(2m+1)^2$}
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 $\sum_{m\geq0}(2m+1)^{-2}=\pi^2/8$.
Back to article page
