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What is the intuition behind the formula for $\pi^4$ that is mentioned on this MathOverflow post https://mathoverflow.net/questions/486961/some-series-related-to-pi4-and-zeta3 ?

What I mean is that I am aware that $\sum_1^\infty \frac{1}{k^4}=\frac{\pi^4}{90}$, and I can see that in Guillera's sum the ratio of polynomial part of each term is also of order $\frac{1}{k^4}$, so somehow the other factors manage to accelarate convergence exactly to $\pi^4$, but it is mysterious to me.

Archie
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  • There is a related question about a series involving $\pi$ and $\sqrt{5}$ by Ramanujan, which has a fine answer on MSE by moderator Paramanand Singh, which involves so-called hypergeometric series based on elliptic integrals, presumably some variant can be worked out for Guillera's formula as well, so not a very intuitive process but still an explanation https://math.stackexchange.com/questions/4898590/how-did-ramanujan-find-sum-n-0-infty-1n-frac1-2-n1-4-n3-4-nn?rq=1 – Archie Feb 03 '25 at 07:04

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