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Sun, in 2022, introduced a conjectured evaluation for a series of convergencerate $\frac{1}{2}$ involving harmonic numbers. We prove both this conjectureand a stronger version of this conjecture, using a summation technique based ona beta-type integral we had previously introduced. Our full proof also requiresapplications 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 formuladue to Chu, Watson's formula for ${}_{3}F_{2}(1)$-series, the Gauss summationtheorem, Euler's formula for ${}_{2}F_{1}$-series, elliptic integral singularvalues, and lemniscate-like constants recently introduced by Campbell and Chu.The techniques involved in our proof are useful, more broadly, in the reductionof difficult sums of convergence rate $\frac{1}{2}$ to previously evaluableexpressions.

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