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arXiv · 2304.00360

On a conjecture on a series of convergence rate $\frac{1}{2}$

John M. Campbell
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Abstract

Sun, in 2022, introduced a conjectured evaluation for a series of convergence rate $\frac{1}{2}$ involving harmonic numbers. We prove both this conjecture and a stronger version of this conjecture, using a summation technique based on a beta-type integral we had previously introduced. Our full proof also requires applications of Bailey's ${}_{2}F_{1}\left( \frac{1}{2} \right)$-formula, Dixon's ${}_{3}F_{2}(1)$-formula, an almost-poised version of Dixon's formula due to Chu, Watson's formula for ${}_{3}F_{2}(1)$-series, the Gauss summation theorem, Euler's formula for ${}_{2}F_{1}$-series, elliptic integral singular values, and lemniscate-like constants recently introduced by Campbell and Chu. The techniques involved in our proof are useful, more broadly, in the reduction of difficult sums of convergence rate $\frac{1}{2}$ to previously evaluable expressions.

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John M. Campbell. 2023-04-01. On a conjecture on a series of convergence rate $\frac{1}{2}$. https://arxiv.org/abs/2304.00360

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