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](https://oeis.org/search?q=3%2C9%2C5%2C3%2C3%2C8%2C5%2C6%2C7%2C3%2C6%2C7%2C4%2C4%2C5%2C5%2C6%2C6%2C0%2C3&language=english&go=Search) recognizes this, but doesn't have much information. This constant is conjectured to be irrational, transcendental and normal. [This](https://math.stackexchange.com/questions/2009336/closed-form-for-prod-i-2-infty-1-frac1i) **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 isn't a function(it is, but the product has to be solved before differentiating). 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. I link to an article with information about this constant will also help.  
 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.