Let $$a_n=\frac{1}{n+\frac{1}{2}}\left [\frac{\Gamma(n)}{\Gamma(n+\frac{1}{2})}\right]^2,$$ and $$b_n=\frac{1}{n^2}.$$ On the ground of hydrogen atom quantum physics, it was shown in http://arxiv.org/abs/1510.07813 (Quantum Mechanical Derivation of the Wallis Formula for $\pi$, by T. Friedmann and C. R. Hagen) that $$\lim\limits_{n\to\infty}\frac{a_n}{b_n}=1.$$ Therefore, since $$\sum\limits_{n=1}^\infty b_n=\frac{\pi^2}{6},$$ the series $\sum\limits_{n=1}^\infty a_n$ is convergent, according to the limit comparison test. Alternatively we can use direct comparison test, because $a_n\le b_n$ for all $n$ (also from physics).

Can the sum $\sum\limits_{n=1}^\infty a_n$ be calculated explicitly?