Few days ago I thought about proof of :$$\frac{1}{3}+\frac{1}{3\cdot 5} + \dots = \sqrt{\frac{e\pi}{2}}$$. I tried to represent my sum as : $$\sum\frac{2n!!}{(2n+1)!}$$, so after I was thinking about Wallis product, but that doesn't give me anything.

May anyone give me a hint what is the best way to start my proof?