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.
Categories: math.NT math.CA