# How can I prove this formula? $\cdots=\sqrt{\frac{e\pi}2}$ [duplicate]

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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?

## marked as duplicate by Bumblebee, egreg, J. M. is a poor mathematician, Hans Lundmark, Community♦Dec 27 '16 at 22:40

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• I don't understand the sum. What do the dots in the middle mean? – lulu Dec 27 '16 at 18:21
• With respect to the continued fraction, see this. – J. M. is a poor mathematician Dec 27 '16 at 18:22
• @lulu I edited to specify what this means. – E. Joseph Dec 27 '16 at 18:24
• Thanks. I was unaware of this formula, but a generalization of it can be found here – lulu Dec 27 '16 at 18:25
• @Sil Thank you, that is exactly what I was looking for! – E. Joseph Dec 27 '16 at 18:27

## 1 Answer

$$S= 1+\frac{1}{1\cdot 3}+\frac{1}{1\cdot 3\cdot 5}+\ldots = \sum_{n\geq 0}\frac{1}{(2n+1)!!}=\sum_{n\geq 0}\frac{2^n n!}{(2n+1)!}$$ can be written (through Euler's Beta function) as $$S = \sum_{n\geq 0}\frac{2^n}{n!}B(n+1,n+1) = \int_{0}^{1}\sum_{n\geq 0}\frac{(2x(1-x))^n}{n!}\,dx$$ or as: $$S = \int_{-1/2}^{1/2}\exp\left[\frac{1}{2}-2x^2\right]\,dx.$$ The remaining part follows from the well-known continued fraction expansion for the (complementary) error function: $$\frac{1}{\sqrt{2\pi}}\int_{z}^{+\infty}e^{-x^2/2}\,dx = \frac{e^{-z^2/2}}{\sqrt{2\pi}}\cdot \frac{1}{z+\frac{1}{z+\frac{2}{z+\frac{3}{\ldots}}}}$$