# Thread: Computing Sums given Fourier Series

1. ## Computing Sums given Fourier Series

Hello,

I am trying to compute the sum:

Summation (1/ (n^2)): boundaries n=1 to infinity

Given that the Fourier series for f(x) = x/2 is
f(x) = summation {[((-1)^(n+1)) / n)]*sinnx

I would really appreciate some help on how to perform these calculations. The text I am using doesn't give examples on these types of problems and I really don't know where to start.

Thanks!!

2. Originally Posted by Chitownmegs
Hello,

I am trying to compute the sum:

Summation (1/ (n^2)): boundaries n=1 to infinity

Given that the Fourier series for f(x) = x/2 is
f(x) = summation {[((-1)^(n+1)) / n)]*sinnx
Let $\displaystyle f( x ) = \frac{x}{2}$ for $\displaystyle -\pi< x < \pi$ and $\displaystyle f(\pi)=f(-\pi)=0$.
Now define $\displaystyle \bar f$ to be the periodic extension of $\displaystyle f$.

Then (as you are given),
$\displaystyle \frac{x}{2} = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\sin (nx)$ for $\displaystyle 0 < x < \pi$

To solve your problem try doing the following: integrate both sides and adjust the correct constant. Then substitute into this new equation $\displaystyle x=\pi/2$ and see if anything happens.

3. ## Response to Hacker

Thanks, Hacker. However, when integrating and substituting, I end up with pi^2/8, when I know the answer is pi^2/ 6...!

What am I doing wrong here?

4. Originally Posted by Chitownmegs
Thanks, Hacker. However, when integrating and substituting, I end up with pi^2/8, when I know the answer is pi^2/ 6...!

What am I doing wrong here?
Must you do it by this series? It is much easier to use the Fourier Series for $\displaystyle x^2$ on $\displaystyle [-1,1]$. And as for why it doesn't work, did you remember to include the constant of integration?