$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series .

Asked Mar 12 '24 at 10:58

Active Mar 13 '24 at 16:02

Viewed 104 times

Conjecture :

$$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/1!+1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$

Where $a_n$ is an integer such that :

$$0\leq a_n\leq n$$

Some arguments :

I think the best we can say is until we have the (says 20 because of floating point ) first term which seems true currently more it goes more we have choice in the $a_n$ integers more the conjecture shall be true .

Another argument : if we stop to $a_k$ and take all the next terms equal to $n$ it seems always greater than the product .

Context :

For a context I was using the Engels form of $\ln 2$ which is near from the value of the product . There is also the Jacobi theta function as start for a formal proof .

Question :

How to (dis)prove the conjecture ?

Second question :

Following the comment what is the factoradic representation starting from $1!$ ?

edited Mar 13 '24 at 16:02

asked Mar 12 '24 at 10:58

Miss and Mister cassoulet char

3,511

2

I guess every real number has such an expansion in the factorial number system. – Gary Mar 12 '24 at 11:29
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See https://en.wikipedia.org/wiki/Factorial_number_system – GEdgar Mar 12 '24 at 12:37
@Gary Thanks for the comment what about the constraint ? – Miss and Mister cassoulet char Mar 13 '24 at 10:49
@MissandMistercassouletchar Yes, in fact you take $0\le a_n < n$. – Milten Mar 13 '24 at 13:44
@Milten Thanks! sometimes it seems so obvious ...for your first comment it's $\leq n$ – Miss and Mister cassoulet char Mar 13 '24 at 16:03
@MissandMistercassouletchar No, I meant you can actually take $0\le a_n\le n-1$. Which is because $\frac n{n!}=\frac 1{(n-1)!}$, so you would increase the previous digit instead. See also https://oeis.org/wiki/Factorial_numeral_system – Milten Mar 14 '24 at 01:33
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This series is from this deleted answer – Тyma Gaidash Mar 14 '24 at 02:14

$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series .

Conjecture :

0 Answers0