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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't answer. Here's it: $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)$$

• This is surely convergent, many tests work. Wolfram Alpha couldn't evaluate it, but gave an approximate value of $$0.395338567367445566032356200431180613$$

• The decimal expansion is OEIS A282529, but the entry doesn't have much information. This constant is conjectured to be irrational, transcendental, and normal.

• This Math.SE question asks specifically for a closed form, but it has no answers, so it doesn't solve my question.

Here's the work I did: \begin{align} \prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3\cdots1\cdot2\cdot\cdots N}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}\cdots(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 of $N$ (it is, but the product should be solved before differentiating).

How can I evaluate it? A link to an article containing information about the 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.
I realized that what I did was not useful. I did some research and found these facts:

• A representation of the Barnes-G function is $$G(N)=\frac{\Gamma(N)^{N-1}}{K(N)}$$ Where $K$ is the K-function.

• A representation of the K-function is $$K(z)=\mathrm{exp}[\zeta'(-1,z)-\zeta'(-1)]$$

Now I used the first point and simplified the product to $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)=\lim_{N\to\infty}\frac{K(N+2)}{\Gamma(N+2)^2}\prod_{k=2}^{N}\frac{k!-1}{(N+1)!}$$ How can this be simplified? Is there any suction related to this? The hard thing to evaluate is this: $$\prod_{k=2}^{N}(k!-1)$$ I looked up in this article but couldn't find a related function. Is there an article that discusses(or at least, mentions) this product?
Now my main question has become:

Simplify, give information about or represent in terms of special function the product:$$\prod_{k=2}^{N}(k!-1)$$

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't answer. Here's it: $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)$$

• This is surely convergent, many tests work. Wolfram Alpha couldn't evaluate it, but gave an approximate value of $$0.395338567367445566032356200431180613$$

• The decimal expansion is OEIS A282529, but the entry doesn't have much information. This constant is conjectured to be irrational, transcendental, and normal.

• This Math.SE question asks specifically for a closed form, but it has no answers, so it doesn't solve my question.

Here's the work I did: \begin{align} \prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3\cdots1\cdot2\cdot\cdots N}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}\cdots(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 of $N$ (it is, but the product should be solved before differentiating).

How can I evaluate it? A link to an article containing information about the 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.
I realized that what I did was not useful. I did some research and found these facts:

• A representation of the Barnes-G function is $$G(N)=\frac{\Gamma(N)^{N-1}}{K(N)}$$ Where $K$ is the K-function.

• A representation of the K-function is $$K(z)=\mathrm{exp}[\zeta'(-1,z)-\zeta'(-1)]$$

Now I used the first point and simplified the product to $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)=\lim_{N\to\infty}\frac{K(N+2)}{\Gamma(N+2)^2}\prod_{k=2}^{N}\frac{k!-1}{(N+1)!}$$ How can this be simplified? Is there any suction related to this? The hard thing to evaluate is this: $$\prod_{k=2}^{N}(k!-1)$$ I looked up in this article but couldn't find a related function. Is there an article that discusses(or at least, mentions) this product?
Now my main question has become:

Simplify, give information about or represent in terms of special functions the product:$$\prod_{k=2}^{N}(k!-1)$$

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't answer. Here's it: $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)$$

• This is surely convergent, many tests work. Wolfram Alpha couldn't evaluate it, but gave an approximate value of $$0.395338567367445566032356200431180613$$

• The decimal expansion is OEIS A282529, but the entry doesn't have much information. This constant is conjectured to be irrational, transcendental, and normal.

• This Math.SE question asks specifically for a closed form, but it has no answers, so it doesn't solve my question.

Here's the work I did: \begin{align} \prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3\cdots1\cdot2\cdot\cdots N}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}\cdots(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 of $N$ (it is, but the product should be solved before differentiating).

How can I evaluate it? A link to an article containing information about the 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.
I realized that what I did was not useful. I did some research and found these facts:

• A representation of the Barnes-G function is $$G(N)=\frac{\Gamma(N)^{N-1}}{K(N)}$$ Where $K$ is the K-function.

• A representation of the K-function is $$K(z)=\mathrm{exp}[\zeta'(-1,z)-\zeta'(-1)]$$

Now I used the first point and simplified the product to $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)=\lim_{N\to\infty}\frac{K(N+2)}{\Gamma(N+2)^2}\prod_{k=2}^{N}\frac{k!-1}{(N+1)!}$$ How can this be simplified? Is there any suction related to this? The hard thing to evaluate is this: $$\prod_{k=2}^{N}(k!-1)$$ I looked up in this article but couldn't find a related function. Is there an article that discusses(or at least, mentions) this product?

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't answer. Here's it: $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)$$

• This is surely convergent, many tests work. Wolfram Alpha couldn't evaluate it, but gave an approximate value of $$0.395338567367445566032356200431180613$$

• The decimal expansion is OEIS A282529, but the entry doesn't have much information. This constant is conjectured to be irrational, transcendental, and normal.

• This Math.SE question asks specifically for a closed form, but it has no answers, so it doesn't solve my question.

Here's the work I did: \begin{align} \prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3\cdots1\cdot2\cdot\cdots N}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}\cdots(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 of $N$ (it is, but the product should be solved before differentiating).

How can I evaluate it? A link to an article containing information about the 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.
I realized that what I did was not useful. I did some research and found these facts:

• A representation of the Barnes-G function is $$G(N)=\frac{\Gamma(N)^{N-1}}{K(N)}$$ Where $K$ is the K-function.

• A representation of the K-function is $$K(z)=\mathrm{exp}[\zeta'(-1,z)-\zeta'(-1)]$$

Now I used the first point and simplified the product to $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)=\lim_{N\to\infty}\frac{K(N+2)}{\Gamma(N+2)^2}\prod_{k=2}^{N}\frac{k!-1}{(N+1)!}$$ How can this be simplified? Is there any suction related to this? The hard thing to evaluate is this: $$\prod_{k=2}^{N}(k!-1)$$ I looked up in this article but couldn't find a related function. Is there an article that discusses(or at least, mentions) this product?
Now my main question has become:

Simplify, give information about or represent in terms of special function the product:$$\prod_{k=2}^{N}(k!-1)$$

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't answer. Here's it: $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)$$

• This is surely convergent, many tests work. Wolfram Alpha couldn't evaluate it, but gave an approximate value of $$0.395338567367445566032356200431180613$$

• The decimal expansion is OEIS A282529, but the entry doesn't have much information. This constant is conjectured to be irrational, transcendental, and normal.

• This Math.SE question asks specifically for a closed form, but it has no answers, so it doesn't solve my question.

Here's the work I did: \begin{align} \prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3\cdots1\cdot2\cdot\cdots N}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}\cdots(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 of $N$ (it is, but the product should be solved before differentiating).

How can I evaluate it? A link to an article containing information about the 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.
I realized that what I did was not useful. I did some research and found these facts:

• A representation of the Barnes-G function is $$G(N)=\frac{\Gamma(N)^{N-1}}{K(N)}$$ Where $K$ is the K-function.

• A representation of the K-function is $$K(z)=\mathrm{exp}[\zeta'(-1,z)-\zeta'(-1)]$$

Now I used the first point and simplified the product to $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)=\lim_{N\to\infty}\frac{K(N+2)}{\Gamma(N+2)^2}\prod_{k=2}^{N}\frac{k!-1}{(N+1)!}$$ How can this be simplified?

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't answer. Here's it: $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)$$

• This is surely convergent, many tests work. Wolfram Alpha couldn't evaluate it, but gave an approximate value of $$0.395338567367445566032356200431180613$$

• The decimal expansion is OEIS A282529, but the entry doesn't have much information. This constant is conjectured to be irrational, transcendental, and normal.

• This Math.SE question asks specifically for a closed form, but it has no answers, so it doesn't solve my question.

Here's the work I did: \begin{align} \prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3\cdots1\cdot2\cdot\cdots N}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}\cdots(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 of $N$ (it is, but the product should be solved before differentiating).

How can I evaluate it? A link to an article containing information about the 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.
I realized that what I did was not useful. I did some research and found these facts:

• A representation of the Barnes-G function is $$G(N)=\frac{\Gamma(N)^{N-1}}{K(N)}$$ Where $K$ is the K-function.

• A representation of the K-function is $$K(z)=\mathrm{exp}[\zeta'(-1,z)-\zeta'(-1)]$$

Now I used the first point and simplified the product to $$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)=\lim_{N\to\infty}\frac{K(N+2)}{\Gamma(N+2)^2}\prod_{k=2}^{N}\frac{k!-1}{(N+1)!}$$ How can this be simplified? Is there any suction related to this? The hard thing to evaluate is this: $$\prod_{k=2}^{N}(k!-1)$$ I looked up in this article but couldn't find a related function. Is there an article that discusses(or at least, mentions) this product?