I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can answer. Here's it: $$\prod_{n=2}^{\infty}1-\frac{1}{n!}$$ This is surely convergent, many tests work. WolframAlpha couldn't evaluate it but gave an approximate value $0.395338567367445566032356200431180613$.
Oeis recognizes this, but doesn't have much information. This constant is conjectured to be irrational, transcendental and normal. This is the same stackexchange question, but it has no answers, so it doesn't solve my question. Here's the work I did: \begin{align} \prod_{n=2}^{\infty}1-\frac{1}{n!}&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3...1\cdot2\cdot...N}\\ &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}...(N-1)^2N^1} \end{align} Now I don't know how to proceed. L'Hopital's rule doesn't work, since the numerator and denominator aren't functions. How can I evaluate it? Is this product a particular value of a special function? If not, can we generalize the product as a function, or can we generalize the numerator and denominator as a function? In that case, L'Hopital's rule can be applied.
Any help would be appreciated.
Note: A closed form isn't necessary; converting the product into a sum or integral will also help. Some special function representations will also be good.