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.
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).