I found in the book *Escapades Arithmétiques* written by Frédéric Laroche this formula:

$$1+\frac 1{1\cdot 3}+\frac 1{1\cdot 3\cdot 5}+\cdots+\frac 1{1+\frac 1{1+\frac 2{1+\frac 3{1+\cdots}}}}=\sqrt{\frac{e\pi}2}.$$

Perhaps in a more explicit way, the first part of this formula is:

$$\sum_{k=0}^\infty \left(\prod_{j=0}^k (2j+1)\right)^{-1}.$$

What I do not like is that this formula (beautiful in my opinion) is written without any proof nor reference.

Do you have an idea on how to prove such a result?

Do you know a book that gives the proof of this formula?