Sum of Half Factorials!

We know that the gamma function $\Gamma (1+x) = x\Gamma {x} = x!$ and $\Gamma (\frac{1}{2}) = \sqrt{\pi}$ . Therefore, the sum:

$\begin{aligned} S & = \frac{1}{(\frac{1}{2})!} + \frac{1}{(\frac{3}{2})!} + \frac{1}{(\frac{5}{2})!} + \frac{1}{(\frac{7}{2})!} + ... \\ & = \frac {1} {\Gamma (\frac{3}{2})} + \frac {1} {\Gamma (\frac{5}{2})} + \frac {1} {\Gamma (\frac{7}{2})} + \frac {1} {\Gamma (\frac{9}{2})} + ... \\ & = \frac {1} {\frac{1}{2} \Gamma (\frac{1}{2})} + \frac {1} {\frac{3}{2} \Gamma (\frac{3}{2})} + \frac {1} {\frac{5}{2} \Gamma (\frac{5}{2})} + \frac {1} {\frac{7}{2} \Gamma (\frac{7}{2})} + ... \\ & = \frac {1} {\frac{1}{2} \Gamma (\frac{1}{2})} + \frac {1} {\frac{3}{2} \dot{} \frac{1}{2} \Gamma (\frac{1}{2})} + \frac {1} {\frac{5}{2} \dot{} \frac{3}{2} \dot{} \frac{1}{2} \Gamma (\frac{1}{2})} + \frac {1} {\frac{7}{2} \dot{} \frac{5}{2} \dot{} \frac{3}{2} \dot{} \frac{1}{2} \Gamma (\frac{1}{2})} +... \\ & = \frac{1}{\sqrt{\pi}} \left( \frac{2}{1} + \frac{2^2}{1\dot{}3} + \frac{2^3}{1\dot{}3\dot{}5} + \frac{2^4}{1\dot{}3\dot{}5\dot{}7} +... \right) \\ & = \frac{1}{\sqrt{\pi}} \sum_{n=0}^\infty {\frac{2^{2n+1}}{(2n+1)!!}} \end{aligned}$

We know that:

$\begin{aligned} erf (x) & = \frac{e^{-x^2}}{\sqrt{\pi}} \sum_{n=0}^\infty {\frac{(2x)^{2n+1}}{(2n+1)!!}} \\ \Rightarrow S & = e \space erf(1) \\ & = 2.7182 \times 0.8427 = \boxed{2.29} \end{aligned}$