sequences and series - Proof of this formula for $sqrt{epi/2}$ and similar formulas.

$$\begin{align} \sqrt{\frac{e\pi}{2}}=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}} \end{align}$$

as seen here.

Is there other series that relate $\pi$ and $e$?

Also, it's possible to rewrite the continued fraction above in terms of known functions/numbers?