• Support PF! Buy your school textbooks, materials and every day products Here!

Partial sum of Fourier Coefficients

  • Thread starter sristi89
  • Start date
  • #1
8
0

Homework Statement



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+...

Homework Equations




N/A

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!

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
nicksauce
Science Advisor
Homework Helper
1,272
5
What do you find to be the fourier series for f(x)?
 
  • #3
Dick
Science Advisor
Homework Helper
26,258
618
Just to echo nicksauce, if you found the fourier series as you said, what is it?
 
  • #4
8
0
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.
 
  • #5
8
0
P.S: and n goes from 1 to infinity
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
618
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.
 
  • #7
8
0
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.
 
  • #8
Dick
Science Advisor
Homework Helper
26,258
618
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...
 
  • #9
Dick
Science Advisor
Homework Helper
26,258
618
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.
 

Related Threads for: Partial sum of Fourier Coefficients

Replies
1
Views
994
Replies
4
Views
1K
Replies
4
Views
867
Replies
0
Views
2K
Replies
0
Views
929
Replies
0
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
1K
Top