91Ó°ÊÓ

In mathematics, Parseval's theorem generally refers to the result that the Fourier transfigure is unitary; approximately, that the sum of the forecourt of a function is equal to the sum of the forecourt of it is transfigure.

Given the series$\frac{1}{3^2}+\frac{1}{{15}^2}+\frac{1}{{35}^2}+...$

Value of the series is to be found using Parseval Theorem and Problem 5.11.

From Problem 5.11, write the function as$f\left(x\right)=\frac{1}{\mathrm{π}}+\frac{1}{2}\sin x-\frac{2}{\mathrm{π}}\underset{n-even}{\overset{∞}{∑}}\frac{\cos \left(nx\right)}{n^2-1}$where$n≠0$

Find the mean value of square of the function

Now, use the Parseval Theorem to get

$\frac{1}{4}=\frac{1}{{\mathrm{π}}^2}+\frac{1}{2}{\left(\frac{1}{2}\right)}^2-\frac{1}{2}\underset{n-even}{\overset{∞}{∑}}{\left(\frac{2}{\mathrm{π}\left(n^2-1\right)}\right)}^2$

Solve further to get

$\underset{n-even}{\overset{∞}{∑}}\frac{1}{{\left(n^2-1\right)}^2}=\frac{{\mathrm{π}}^2}{16}-\frac{1}{2}$

Therefore, the sum of series$\frac{1}{3^2}+\frac{1}{{15}^2}+\frac{1}{{35}^2}+...$ is$\frac{{\mathrm{π}}^2}{16}-\frac{1}{2}$ by the use of Parseval’s theorem.