The Fourier series of the function,
$\begin{array}{rll} f(x) & =0, & -\pi < x \leq 0 \\ {} & =\pi – x, & 0 < x < \pi \end{array}$
in the interval $[- \pi, \pi ]$ is
$f(x) = \dfrac{\pi}{4} + \dfrac{2}{\pi} \bigg[ \dfrac{\cos x}{1^2} + \dfrac{\cos 3x}{3^2} + \dots \dots \dots \bigg] + \bigg[ \dfrac{\sin x}{1} + \dfrac{\sin 2x}{2} + \dfrac{\sin 3x}{3} + \dots \dots \dots \bigg].$
The convergence of the above Fourier series at $x=0$ gives
- $\Sigma_{n-1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6} \\$
- $\Sigma_{n-1}^{\infty} \dfrac{(-1)^{n+1}}{n^2} = \dfrac{\pi^2}{12} \\$
- $\Sigma_{n-1}^{\infty} \dfrac{1}{(2n-1)^2} = \dfrac{\pi^2}{8} \\$
- $\Sigma_{n-1}^{\infty} \dfrac{(-1)^{n+1}}{2n-1} = \dfrac{\pi}{4}$