Partial sum of Fourier Coefficients

sristi89

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

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

x^2 0<x<pi

Find the fourier series and use it to show that

(pi^2)/6=1+1/2^2+1/3^2+...

2. Relevant equations


N/A

3. The attempt at a solution

I was able to find the fourier series and my answer matched with the back of the book. But I don't understand how I am supposed to use it to prove the expansion of pi^2/6 . I tried plugging in (pi^2)/6 for f(x) but that didn't work out.


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



2. Relevant equations



3. The attempt at a solution

nicksauce

What do you find to be the fourier series for f(x)?

Dick

Just to echo nicksauce, if you found the fourier series as you said, what is it?

sristi89

I didn't want to type out the fourier series since I am really new to latex but here it is

f(x) =pi^2/6 +[tex]\Sigma[/tex] { (2(-1)^n/n^2) cos nx + ((-1)^(n+1) pi/n + 2/(pi * n^3) [(-1)^n-1] sin nx }

I hope that made sense and that I didn't make any typos.

sristi89

P.S: and n goes from 1 to infinity

Dick

Thanks. Since the mean value of the function over the interval is pi^2/6, I was hoping I could just say put x=0. But you've got that alternating sign and factor of 2 in the cosine part. Are you sure of the series? Otherwise, I'll have to work it out. Didn't want to do that.

sristi89

I am pretty sure about the series. I verified with the back of the book. We were also supposed to show pi^2/12=1-1/2^2+ etc. I plugged in x=o and was able to prove it.

For the pi^2/6, I tried setting x to be pi/2 so that the cosine term goes away but that didn't work out.

Dick

It looks to me like you should be putting x=pi. That turns cos(nx) into (-1)^n makes the sin(nx) vanish. I'm trying to check your series. I get something a lot like it. But I can't seem to get all the parts quite right. Guess I'm not very good at this...

Dick

Doh. I've been being stupid. Regarded as a periodic function f(x) is discontinuous at x=pi. The series doesn't converge to f(pi). It converges to (f(pi)+f(-pi))/2. Or pi^2/2. Do you know why? Now try x=pi in your series.