On the OEIS Wiki immediately after the formula $$\sqrt{\frac{\pi e}{2}}=\frac{1}{1+\mathrm{K}_{i=1}^{\infty}{\frac{i}{1}}}+\sum_{n=0}^{\infty}{\frac{1}{(2n+1)!!}}$$ (where I am using $\mathrm{K}$ as in the third notation here for continued fractions) it is written "Since it is not known whether $\pi e$ is irrational or not, it is thus not known whether $\sqrt{\pi e/2}$ is transcendental or not (although it is obviously irrational)."

How can the irrationality of $\sqrt{\pi e/2}$ be derived from the formula?