試用！

$バーゼル問題の証明例$

$フーリエ級数 {\displaystyle x^2=\frac{{\pi }^2}{3}+\sum _{n=1}^{\mathrm{\infty }}((-1{)}^n\frac{4}{n^2}\cos nx) x\in (-\pi ;\pi )}$

$\begin{array}{rl}{\displaystyle {\pi }^2}& =\frac{{\pi }^2}{3}+\sum _{n=1}^{\mathrm{\infty }}((-1{)}^n\frac{4}{n^2}\cos \pi x) \\ & =\frac{{\pi }^2}{3}+4\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}\\ \frac{2{\pi }^2}{3}& =4\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}\\ \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}& =\frac{{\pi }^2}{6}\end{array}$