Homework Help: Show that this series sums to value shown using Fourier technique

1. Nov 6, 2012

bossman007

1. The problem statement, all variables and given/known data

Use the fourier series technique to show that the following series sums to :
$$1+\frac{1}{3^2}+\frac{1}{5^2}+...=\frac{\pi^2}{8}$$

2. Relevant equations

3. The attempt at a solution

Don't know what the first few steps are...but I assume that I need to first express the sum as $$\frac{1}{(2n-1)^2}$$, but have no idea where to go from there. I know that fourier series involve finding the coefficients for the $$a_0$$, $$a_n$$, and $$b_n$$.

2. Nov 6, 2012

bossman007

it seems the most difficult thing is picking the f(x) function

3. Nov 6, 2012

vela

Staff Emeritus
You need to be familiar with the Fourier series for common functions. I'd look for one that has only odd harmonics.

4. Nov 6, 2012

bossman007

would you be referring to a square wave? if so, i'm having trouble setting it up in relation to this series or whatever I have to do

5. Nov 6, 2012

vela

Staff Emeritus
The square wave doesn't quite do it because you need n2 in the denominator whereas the Fourier coefficients for the square wave are proportional to 1/n, but it's a good place to start.

What can you do to the series to get another factor of n in the denominator?

6. Nov 6, 2012

bossman007

multiply the series by the integral of (cos(n*pi*x)? because taking the integral will yield a 1/n multiplied by the series? , im still confused what to do after I have that

7. Nov 6, 2012

bossman007

If thats completely wrong i apologize, i've had my head buried in fourier series all day and i'm probably mixing stuff up :/

8. Nov 6, 2012

vela

Staff Emeritus
I'm not sure exactly what you mean, but I think you're on the right track. Can you show what you mean using math starting with the Fourier series for the square wave?

9. Nov 6, 2012

bossman007

My classmates led me to believe that i should choose the function for a square wave :

f(x) = -1 when -pi < x < 0
1 when 0 < x < pi

doing this I get $$a_0$$=0

and $$b_n$$ = $$\frac{1}{\pi}\int^\pi_0 sin(n*\pi*x)\,dx$$ , but I dont see how this helps. I know the fact $$cos(n\pi)=(-1)^n$$ and sin(n\pi)=0 pop up in a lot of these problems we have been doing, and I dont see how the square wave b_n coefficient I wrote out makes and sense for this series

10. Nov 6, 2012

vela

Staff Emeritus
You don't want the factor of $\pi$ in the argument of the sine. So what did you get when you did the integral? What happens when n is even? What happens when n is odd?

11. Nov 6, 2012

bossman007

when n is even or odd sin dissapears. doing the integral without the factor of pi in the argument of sine yields:

$$\frac{1}{n\pi}-\frac{cos(n\pi}{n\pi}=\frac{1}{n\pi}-\frac{-1^n}{n\pi}$$

12. Nov 6, 2012

vela

Staff Emeritus
From your final result, if n is even, you have $\cos n\pi = 1$, so bn=0. If n is odd, you get $b_n = \frac{2}{n\pi}$, right? So the first few terms of the series are
$$\frac{2}{\pi}\left(\frac{1}{1}\sin x + \frac{1}{3} \sin 3x + \frac{1}{5} \sin 5x + \cdots \right)$$ Now what do you get if you integrate that?

13. Nov 6, 2012

bossman007

IT GIVES ME MY ORIGINAL SERIES :D, thank u so much, but not sure what I need to do after that to prove it sums to pi^2/8

14. Nov 6, 2012

vela

Staff Emeritus
You have
$$f(x) = \frac{2}{\pi}\left(\frac{1}{1}\sin x + \frac{1}{3} \sin 3x + \frac{1}{5} \sin 5x + \cdots \right)$$ You integrated the righthand side, so you have to integrate the lefthand side. Then picking an appropriate value of x, you'll get the result you desire.

15. Nov 7, 2012

bossman007

many thanks,

little confused on the integral on the left hand and right hand side should be

I have "integral of f(x)" = (2[-cos(x)-(1/(3^2))cos(3x)-1/(5^2))cos(5x)]) / pi

I cant find a value of x that makes it pi^2/8

I know if x=pi, the RHS will yield my initial series multipled by the 2 out front of the brackets, but I'm tripping myself up over the left hand side and what to integrate

I'm also confused because the RHS is infinite series and not sure what to do

Last edited: Nov 7, 2012
16. Nov 7, 2012

vela

Staff Emeritus
You wrote what f(x) is above, in post 9. You can either use definite integrals with the limits conveniently chosen, or you can use indefinite integrals and deal with the constants of integration.

Also, I just noticed you're missing a factor of 2 in your formula for bn.

17. Nov 7, 2012

bossman007

I ended up getting a close answer to pi^2/8

still not totally sure what my fourier formula should look like with everything plugged int