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!

  • 1
    $\begingroup$ Read some of the answer including mine here $\endgroup$ – user17762 Apr 1 '13 at 20:50
  • $\begingroup$ In case you want a different method, find the Fourier Series of $x^4$, periodic on $[-\pi,\pi]$, and evaluate at $x=\pi$. $\endgroup$ – Fly by Night Apr 1 '13 at 21:02