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?



The sum of an hydrogen atom related infinite series
Using Christian Krattenthaler's [hyp.m](http://www.mat.univie.ac.at/~kratt/hyp_hypq/hyp.html#HYP) tells you the following - the important information is at the end, and the result agrees with Johannes' comment.



```
In[1]:= <<hyp.m

Out[1]= ▒

In[10]:= S = SUM[1/(n+1/2)*(Gamma[n]^2/Gamma[n+1/2]^2),{n,1,Infinity}]

         Infinity
          -----                2
           \           Gamma[n]
Out[10]=    >    ---------------------
           /      1            1     2
          -----  (- + n) Gamma[- + n]
           n=1    2            2

In[19]:= SF = S/.SUMF

              [ 1, 1, 1    ]
              |            |      2
         2  F |  5  3  ; 1 | Ga(1)
           3 2|  -, -      |
              [  2  2      ]
Out[19]= --------------------------
                       3 2
                  3 Ga(-)
                       2

In[21]:= SF/.SListe

Be sure to apply "FOrdne" before using the following information!

                        2
           2 S3261 Ga(1)
Out[21]= {{--------------}}
                   3 2
              3 Ga(-)
                   2

In[22]:= SF/.S3261

                         1     1     1   3                              3    1     1     1   3
                     [ -(-), -(-), -(-), - ]      [   1     1     ]   [ -, -(-), -(-), -(-), - ]
                2    |   2     2     2   2 |      | -(-), -(-)    |   | 2    2     2     2   2 |
         2 Ga(1)  (Ga|                     | -  F |   2     2 ; 1 | Ga|                        |)
                     |      3   1  1  1    |   2 1|               |   |      3                 |
                     [    -(-), -, -, -    ]      [     0         ]   [    -(-), 1, 1, 1, 0    ]
                            2   2  2  2                                      2
Out[22]= ----------------------------------------------------------------------------------------
                                                      3 2
                                                 3 Ga(-)
                                                      2

In[23]:= SF/.S3261/.SListe

Be sure to apply "FOrdne" before using the following information!

                           1     1     1   3               3    1     1     1   3
                       [ -(-), -(-), -(-), - ]           [ -, -(-), -(-), -(-), - ]
                  2    |   2     2     2   2 |           | 2    2     2     2   2 |
           2 Ga(1)  (Ga|                     | - S2103 Ga|                        |)
                       |      3   1  1  1    |           |      3                 |
                       [    -(-), -, -, -    ]           [    -(-), 1, 1, 1, 0    ]
                              2   2  2  2                       2
Out[23]= {{-------------------------------------------------------------------------}}
                                                3 2
                                           3 Ga(-)
                                                2

In[33]:= R = SF/.S3261/.S2103/.Gzerl

                        1  3    3            1  3    3 2
                   Ga(-(-))  Ga(-)      Ga(-(-))  Ga(-)
                2       2       2            2       2
         2 Ga(1)  (--------------- - ----------------------)
                        3      1 3        3      1 2      2
                   Ga(-(-)) Ga(-)    Ga(-(-)) Ga(-)  Ga(1)
                        2      2          2      2
Out[33]= ---------------------------------------------------
                                   3 2
                              3 Ga(-)
                                   2

In[42]:= Simplify[R]

                1  3       2      1     3
         2 Ga(-(-))  (Ga(1)  - Ga(-) Ga(-))
                2                 2     2
Out[42]= ----------------------------------
                     3      1 3    3
              3 Ga(-(-)) Ga(-)  Ga(-)
                     2      2      2

In[52]:= ?S3261
Summation formula (Slater, Appendix (III.31)) in form of a rule.

In[53]:= ?S2103
Summation formula (Slater, Appendix (III.3)) in form of a rule.

```

