Are there any nice ways to prove that ζ(4)=∞∑n=11n4=π490?
I already know some proofs that give all values of ζ(n) for positive even integers n (like #7 on Robin Chapman’s list or Qiaochu Yuan’s answer in the linked question). I’m not so much interested in those kinds of proofs as I am those that are specifically for ζ(4).
I would be particularly interested in a proof that isn’t an adaption of one that ζ(2)=π2/6.
In the same spirit of the 1st proof of this answer. If we substitute π for x in the Fourier trigonometric series expansion of f(x)=x4, with −π≤x≤π,