Summing an infinite series

1. Apr 18, 2016

spaghetti3451

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

Show that $\sum_{n=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}$.

2. Relevant equations

3. The attempt at a solution

$\frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots$.

Do I now factorise?

2. Apr 18, 2016

Fightfish

No, I'm pretty sure there's no way to directly perform the summation in this form.

You can either make use of the integral form of the Riemann zeta function or a neat trick using Fourier series (Parseval's theorem).

3. Apr 18, 2016

Ray Vickson

$$\frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots$$
is wrong. The only time it could be correct is if $n = 1$ and you include only one term on the right-hand-side.

The solution to your problem cannot involve just pre-calculus methods, but instead, very likey involves advanced methods in calculus that use matrrial beyond that found in first or second courses in calculus.

4. Apr 18, 2016

spaghetti3451

A typo!

5. Apr 18, 2016

Ray Vickson

OK, but the rest of my answer applies unchanged.

Last edited by a moderator: Apr 19, 2016