Nice proofs of ζ(4)=π490\zeta(4) = \frac{\pi^4}{90}?

I know some nice ways to prove that ζ(2)=n=11n2=π2/6. For example, see Robin Chapman’s list or the answers to the question “Different methods to compute n=11n2?”

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.

Answer

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 πxπ,

x4=15π4+n=18n2π248n4cosnπcosnx,

we obtain

π4=15π4+n=18n2π248n4cos2nπ=15π4+8π2n=11n248n=11n4.

Hence

n=11n4=π448(1+15+86)=π448815=190π4.

Attribution
Source : Link , Question Author : Mike Spivey , Answer Author : Community

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