The sum of the reciprocals of fourth powers

Asked Apr 01 '13 at 20:47

Active Apr 01 '13 at 20:47

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This problem is an extension of the well known basel problem and involves finding the sum of

1 + 1/16 + 1/81 ... = 1/1^4 + 1/2^4 + 1/3^4 ... 1/n^4 where n tends to infinity

Euler managed to prove that the sum is finite and converges to pi^4/90. What I do not understand is how he managed to accomplish this.

I have read his proof of the problem involving 2nd powers and fully understand the logic but I fail to see how he manages to extend that to this problem (I have yet to find a complete proof online).

Would somebody please explain the method? Does it involve manipulating the second powers or does it involve some other type of derivation?

Thanks Ahead of Time!

asked Apr 01 '13 at 20:47

Sidharth Ghoshal

16,771

1

Read some of the answer including mine here – Apr 01 '13 at 20:50
1

In case you want a different method, find the Fourier Series of $x^4$, periodic on $[-\pi,\pi]$, and evaluate at $x=\pi$. – Fly by Night Apr 01 '13 at 21:02

The sum of the reciprocals of fourth powers

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