Solve Parseval's Identity for 1+\frac{1}{9}+\frac{1}{25}+\cdot\cdot\cdot

[TheFerruccio]

Homework Statement

Obtain the result of the infinite sum [tex]1+\frac{1}{9}+\frac{1}{25}+\cdot\cdot\cdot[/tex]

By applying Parseval's Identity to the Fourier series expansion of
0 if [tex]-\frac{\pi}{2} < x < \frac{\pi}{2}[/tex]
1 if [tex]\frac{\pi}{2} < x < \frac{3\pi}{2}[/tex]

Homework Equations

[tex]2a_0^2+\sum_n{(a_n^2+b_n^2)}\leq\frac{1}{\pi}\int_{-\pi}^{\pi}\! f^2(x) \, \mathrm{d}x.[/tex]

The Attempt at a Solution

I got the solution to the Fourier series, and I know it's correct.

The terms for [tex]a_n[/tex] in the Fourier series expansion are [tex]\frac{-2}{n\pi}[/tex] if n=1,5,9,13,..., and [tex]\frac{2}{n\pi}[/tex] if n=3,7,11,15,...
The [tex]b_n[/tex] terms are 0, since it is an even function about 0.

I'm just not sure how to use this information in Parseval's Identity.

 

[lanedance]

so write your sum out and see if you can relate it to the given series

in the infinite limit the inequality becomes equality, and the integral should be easy to evaluate