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Question. I wonder whether there exists a closed form for the following infinite product $$ \prod_{i=2}^{\infty} (1 - \frac{1}{i!}) $$

I can prove that the product is convergent, but failed to attain a closed form without luck. Any hint is really appreciated.

PSPACEhard
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  • This looks like a good question to me. I don't understand the close votes. Yes, the OP hasn't got any working to show, but that is because the question is difficult. (OP, perhaps you could tell us where this question came from?) – TonyK Nov 11 '16 at 15:21
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    $$\sum_{2\leq i<j}\frac{1}{i!j!}=\frac{1}{2}\left[\left(\sum_{n\geq 2}\frac{1}{n!}\right)^2-\sum_{n\geq 2}\frac{1}{n!^2}\right]=1+\frac{(e-2)^2-I_0(2)}{2}$$ – Jack D'Aurizio Nov 11 '16 at 16:37
  • You may approach $$\sum_{2\leq i<j<k}\frac{1}{i!j!k!}$$ and so on in a similar fashion, then collect back such contributes. – Jack D'Aurizio Nov 11 '16 at 16:38
  • @TonyK Thanks. I came up with this problem when I found the following question. That question has a very nice solution, so I wonder whether we can attain a closed form too if the term $(1 - \frac{1}{n^2})$ is changed to $(1 - \frac{1}{n!})$. – PSPACEhard Nov 12 '16 at 05:02
  • @JackD'Aurizio Thanks. I will try your approach. – PSPACEhard Nov 12 '16 at 05:04
  • http://math.stackexchange.com/questions/460579/how-to-compute-prod-n-1-infty-left1-frac1n-right?rq=1 – Martin Nicholson Nov 13 '16 at 20:44

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This question might help you. It has the same question. I don't think that this product has a closed form. This is the OEIS sequence, but it doesn't have much information. Not much is known about this constant. It is conjectured to be normal, irrational and transcendental. I think this is probably irrational, but I don't know on what basis it is conjectured to be normal. The product can be converted to: $$\prod_{i=2}^{\infty}\left(1-\frac{1}{n!}\right)=e^{-\sum_{n=2}^{\infty} \sum_{k=1}^{\infty} \frac{1}{k (n!) ^ k}}$$ The question is active, it might get answers(the question is asked by me).

  • This doesn't answer the question; it is more appropriate as a comment. (You needn't repeat so much of the content of your version of the question, since you've provided a direct link to it.) BTW, since you have re-posted your Math.SE question to MathOverflow, you should delete the former. (At the very least, put the "I moved this question" notification at the top of the body; comments are easily overlooked and could be obscured. That said, deleting an abandoned question is better. (If you had received answers, then deleting would be inconsiderate to answerers, but that's not the case here.)) – Blue Oct 24 '20 at 09:20