- #1

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## 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?

- Thread starter spaghetti3451
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- #1

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

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

Do I now factorise?

- #2

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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

Ray Vickson

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Your "equation"## 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?

[tex] \frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots [/tex]

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

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A typo!Your "equation"

[tex] \frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots [/tex]

is wrong. The only time it could be correct is if ##n = 1## and you include only one term on the right-hand-side.

- #5

Ray Vickson

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OK, but the rest of my answer applies unchanged.A typo!

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