<s>[INST] Hypergeometric sum specific value How to show?
$${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$
It numerically is very close, came up when evaluating:
$$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 \times 3}{1 \times 3 \times 5} + \frac{1 \times 2 \times 3 \times 4}{1 \times 3 \times 5 \times 7} + \cdots = 1 + \frac{\pi}{2} ~ ?$$
I am missing something trivial, I am sure.
 [/INST] You may use the general formula from Brychkov, Prudnikov, Marichev, Integral and Series, Vol.3:
$$
_{2}F_{1}(1,1;\frac{1}{2};x)=(1-x)^{-1}\left(1+\frac{\sqrt{x}\arcsin{\sqrt{x}}}{\sqrt{1-x}}\right)
$$
and put in it $x=\frac{1}{2}$.
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