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Q: Convergence acceleration technique for $\zeta(4)$ (or for $\eta(4)$) via creative telescoping?

Américo Tavares Question. Is it already known whether the $\zeta(4)$ accelerated convergence series $$\zeta(4)=\frac{36}{17}\sum_{1}^{\infty } \frac{1}{n^{4}\binom{2n}{n}},\tag{1}$$ proved for instance in [1, Corollaire 5.3], could be obtained by a similar technique to those described in [2, ยง1] for $\z...


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