parseval's theorem

**PvtBillPilgrim** · January 19th 2009, 05:15 PM

How can I use Parseval's theorem to find the summation from n =1 to infinity of (1/n^4)?
I'm pretty sure I can calculate the Fourier cosine or sine series for a function; however, I'm not sure what this function should be. I tried f(x) = x, but I get something with (-1)^n - 1 in the numerator with n^2 in the denominator. Because it has this oscillating part, I can't solve the summation.
Tough to explain I guess, but does anyone have an idea what function to use and how you do it? I've been trying to get 1/n^2 in the denominator of the an terms (and then you square it), but I can't get it with just constants. I know the answer is pi/90. I would appreciate any help.

**Opalg** · January 20th 2009, 12:10 AM

Try the function f(x) = x^2 on the interval (–π,π). You should find that $a_0 = 2\pi^2/3$ , and $a_n = 4(-1)^n/n^2$ when n≠0. Then Parseval's theorem gives $\sum_{n=1}^\infty\frac1{n^4} = \frac{\pi^4}{90}$ .

**PvtBillPilgrim** · January 21st 2009, 01:49 PM

Thank you very much. This does work.

What function do I pick for the summation from n =1 to infinity of (1/n^6)? If I try a Fourier cosine series with x^3, it seems I get the oscillation problem again.

Any help appreciated.

**PvtBillPilgrim** · January 22nd 2009, 11:23 AM

Does anyone know a function to use, please? I can't get 1/n^3 in the series with easy to use terms in the numerator.

**Mathstud28** · January 22nd 2009, 11:47 AM

Originally Posted by PvtBillPilgrim

Does anyone know a function to use, please? I can't get 1/n^3 in the series with easy to use terms in the numerator.

If you can you should let someone know...because then you would have found $\zeta(3)$ (Apery's constant) whose irrationality was only found a couple of years ago...the actual value is unknown.

Here is an article if you are interested

http://mathworld.wolfram.com/AperysConstant.html

**PvtBillPilgrim** · January 22nd 2009, 12:07 PM

Well, I want to determine the summation 1/n^6. How do I do this? I believe it's pi^6/(956).

**Opalg** · January 22nd 2009, 12:34 PM

Originally Posted by Mathstud28

If you can you should let someone know...because then you would have found $\zeta(3)$ (Apery's constant) whose irrationality was only found a couple of years ago...the actual value is unknown.

... and yet it clearly states here that "The Riemann zeta function $\zeta(2n)$ may be computed analytically for even n using either contour integration or Parseval's theorem with the appropriate Fourier series" (my underlining). Unfortunately, it doesn't tell you which Fourier series to use.

My guess is that there must be a function whose Fourier coefficients are $(-1)^n/n^3$ . That would get round the problem of having to avoid anything that looks like the Apéry constant, yet still allows you to get ${\textstyle\sum} 1/n^6$ using Parseval's theorem. Have you tried the function $|x^3|$ ? (That's just a guess — I have no idea whether it will work.)

Originally Posted by PvtBillPilgrim

Well, I want to determine the summation 1/n^6. How do I do this? I believe it's pi^6/(956).

Actually $\pi^6/945$ (see the above link).

**magus** · May 2nd 2010, 12:55 AM

Originally Posted by Opalg

... and yet it clearly states here that "The Riemann zeta function $\zeta(2n)$ may be computed analytically for even n using either contour integration or Parseval's theorem with the appropriate Fourier series" (my underlining). Unfortunately, it doesn't tell you which Fourier series to use.

My guess is that there must be a function whose Fourier coefficients are $(-1)^n/n^3$ . That would get round the problem of having to avoid anything that looks like the Apéry constant, yet still allows you to get ${\textstyle\sum} 1/n^6$ using Parseval's theorem. Have you tried the function $|x^3|$ ? (That's just a guess — I have no idea whether it will work.)

Actually $\pi^6/945$ (see the above link).

My text on PDEs that I'm working on has a similar problem but gives the Fourier expansion:

$\dfrac{x}{2} = \sum_{n=1}^{inf}\dfrac{(-1)^{n+1}}{n}\sin{nx}$

I think I know how to start but I'd also like some guidance.

EDIT: N/M That was just to find $\zeta(2)$ You need Bernoulli numbers for a generic formula.

Thread: parseval's theorem

LinkBack

Thread Tools

Display

parseval's theorem

Similar Math Help Forum Discussions

[SOLVED] Computing a sum using Parseval's theorem

Parseval's Theorem

Proving Parseval's Theorem

parseval law

Parseval’s theorem / Fourier series

Search Tags