TITLE: A closed form of $\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$
QUESTION [38 upvotes]: I am looking for a closed form of the following series

\begin{equation}
\mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)
\end{equation}

I have no idea how to answer this question. Wolfram Alpha gives me result:

$$\mathcal{I}\approx2.7415567780803776$$

Could anyone here please help me to obtain the closed form of the series preferably (if possible) with elementary ways (high school methods)? Any help would be greatly appreciated. Thank you.

REPLY [6 votes]: Here is a closed form

$$ \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)x^k= 2 \arcsin \left( x/2 \right)  \left(\pi - \arcsin \left( x/2\right)  \right) .$$

Now just plug in $x=1$ and the result follows.