Convergence of Infinite Series: Solving for the Sum of 1/n^4

[spaghetti3451]

Homework Statement

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

Homework Equations

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?

 

[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).

 

[Ray Vickson]

Homework Statement

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

Homework Equations

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?

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

 

[spaghetti3451]

Your "equation"
$$\frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots$$
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]

A typo!

OK, but the rest of my answer applies unchanged.