# Summing an infinite series

## Homework Statement

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

## 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?

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).

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

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

## 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?

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

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

A typo!

Ray Vickson