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.
Try the function f(x) = x^2 on the interval (–π,π). You should find that , and when n≠0. Then Parseval's theorem gives .
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.
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 (Apery's constant) whose irrationality was only found a couple of years ago...the actual value is unknown.
Originally Posted by PvtBillPilgrim
Here is an article if you are interested
Well, I want to determine the summation 1/n^6. How do I do this? I believe it's pi^6/(956).
My text on PDEs that I'm working on has a similar problem but gives the Fourier expansion:
Originally Posted by Opalg
I think I know how to start but I'd also like some guidance.
EDIT: N/M That was just to find You need Bernoulli numbers for a generic formula.