- #1
buffordboy23
- 548
- 2
Homework Statement
Let f(x) be defined by the following Fourier series for [tex] \left|x\right|[/tex]:
[tex] f(x) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{1,3,...}\frac{cos\left(nx\right)}{n^{2}}[/tex]
Show that
[tex] \sum_{1,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{8}[/tex]
and
[tex] \sum_{1,2,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}[/tex]
The Attempt at a Solution
I was able to find the first sum by letting x = 0. I don't now how to approach the second part since the sum consists of the odd and even n integers, but the Fourier series is only comprised of the odd integers.