Sum of series (quotient of factorials)

Asked Apr 13 '23 at 14:27

Active Apr 13 '23 at 14:50

Viewed 62 times

Does somebody know how to sum this series

$$ \sum_{k=1}^{\infty} \frac{k!}{(2k+1)!!} $$

This problem is from an article of Joseph Blitzstein where he stopped here. The problem asks to find the probability of having 2 black balls and 1 red. Take randomly one, if you take the black it ends, otherwise you put it back and add another black and another red. It is asked to know the expectancy of the random variable

edited Apr 13 '23 at 14:50

asked Apr 13 '23 at 14:27

Victor Esteban

What have you tried? – Nobody Apr 13 '23 at 14:29
I tried to proof it to be the Euler-Mascheroni constant but it is a bit shorter than that. – Victor Esteban Apr 13 '23 at 14:32
Please see this article on MathSE protocol. As onerous as the article may appear to you, it provides a defense mechanism against the MathSE forum being used as a do my homework forum. In particular, please see the Edit-Tools section of the article, and the portion of the article that discusses showing work. It is irrelevant whether the problem is homework. What counts is whether the protocol is observed. – user2661923 Apr 13 '23 at 14:33
The posting's use of double factorial is explained here. – user2661923 Apr 13 '23 at 14:34
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https://math.stackexchange.com/questions/2149340/how-to-prove-this-pi-formula?noredirect=1 – Svyatoslav Apr 13 '23 at 15:17

Sum of series (quotient of factorials)

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