偶数の二重階乗(double factorial / semifactorial)の逆数(reciprocal)の和(無限級数(infinite series))

$$\begin{array}{rcl} \frac{1}{0!!}+\frac{1}{2!!}+\frac{1}{4!!}+\cdots &=&\displaystyle \sum_{k=0}^{\infty}\frac{1}{2^k k!} \\&&\;\dots\;n!!(二重階乗) \neq (n!)! (=階乗凾数の二回反復),\;偶数nの二重階乗n!!=\prod_{k=0}^{\frac{n}{2}}(2k),偶数2k(k\geq0)の二重階乗(2k)!!=2^kk! \\&=&\displaystyle \sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}\right)^k}{k!} \\&=&\displaystyle \mathrm{e}^{\frac{1}{2}} \,\dotso\, \href{https://shikitenkai.blogspot.com/2019/07/blog-post.html}{\sum_{k=0}^{\infty}\frac{x^k}{k!}= \mathrm{e}^x}\\ &=&\displaystyle \sqrt{\mathrm{e}}\\ \end{array}$$