According to Parseval's identity it holds that
![{\displaystyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}=\int _{0}^{1}|f(x)|^{2}dx}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/1178af1e09e4f7a8f50dbf4e657b0a76820a43aa)
where cn are the Fourier coefficients of ƒ(x). In part (b) we calculated the right hand side and found that the above equals to 1/3.
In part (a) we found that the Fourier coefficients are
and
. Hence
![{\displaystyle {\begin{aligned}{\frac {1}{3}}&=\sum _{n=-\infty }^{\infty }|c_{n}|^{2}\\&=|c_{0}|^{2}+2\sum _{n=1}^{\infty }\left|c_{n}\right|^{2}\\&=\left|{\frac {1}{2}}\right|+2\sum _{n=1}^{\infty }\left|-{\frac {1}{2\pi in}}\right|^{2}\\&={\frac {1}{4}}+{\frac {1}{2\pi ^{2}}}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/2fd3c5972227f1c2683f2ea0cde774bed1b595ce)
where we have noted that
in splitting up the sum. Solving for the remaining series yields the final answer
![{\displaystyle \sum _{n=1}^{\infty }n^{-2}={\frac {\pi ^{2}}{6}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/237ff46e319dc0b4953f09112bc398b4b13d6d04)