<s>[INST] Summing the series $ \frac{1}{2n+1} + \frac{1}{2} \cdot \frac{1}{2n+3} + \cdots \ \text{ad inf}$ How does one sum the given series: $$ \frac{1}{2n+1} + \frac{1}{2} \cdot \frac{1}{2n+3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{1}{2n+5} + \frac{ 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{1}{2n+7} + \cdots \ \text{ad inf}$$
Given, such a series, how does one go about solving it. Getting an Integral Representation seems tough for me.
I thought of going along these lines, Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$ but couldn't succeed.
 [/INST] FWIW, Mathematica is able to sum this:
In[1]:= Sum[1/(2 n + 2 k + 1) (2 k - 1)!!/(2 k)!!, {k, 0, \[Infinity]}]

                        3 + 2 n
        Sqrt[Pi] Gamma[-------]
                           2
Out[1]= -----------------------
        (1 + 2 n) Gamma[1 + n]

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