How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?

Question

Does

$$p=\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$$

have any closed form in terms of known mathematical constants? The computer says $$p=3.682154\dots$$ but I don't even know how do devise the converging upper and lower bounds to obtain this result.

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edit Jan. 15: I've got rid of the infinite product in favor of an fastly converging infinite sum over finite products here.

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Thoughts:

$$p=\lim_{n\to \infty}p_n\hspace{.7cm}\text{where}\hspace{.7cm} p_n=p_{n-1}\cdot \left(1+\frac{1}{n!}\right)\hspace{.7cm}\text{with}\hspace{.7cm} p_1=2.$$

So I looked for an emerging pattern

$p_1=(1+\frac{1}{1!})$

$p_2=(1+\frac{1}{1!})(1+\frac{1}{2!})=(1+\frac{1}{1!}+\frac{1}{2!})+(\frac{1}{1!2!})$

$p_3=((1+\frac{1}{1!}+\frac{1}{2!})+\frac{1}{1!2!})(1+\frac{1}{3!}) =(1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!})+(\frac{1}{1!2!}+\frac{1}{1!3!}+\frac{1}{2!3!})+\frac{1}{1!2!3!}$

It appears that

$$p=1+\sum_{n=1}^\infty\sum_{m=1}^\infty a_{nm}$$

where $a_{1m}$ is the sum of terms with one inverse $\frac{1}{m!}$, and then $a_{2m}$ is the sum of (sums of) terms with two inverses $\frac{1}{r!s!}$. For example the term $\frac{1}{1!3!}$ is in the sum, and so I guess I need all the partitions into $n$ numbers. However, we don't want to count $\frac{1}{2!2!}$ and so it's more complicated. I guess the product can be written as a sum of term $(e-1)^n$ minus something, as for example

$(e-1)^2 = \left(\frac{1}{1!} + \frac{1}{2!}+ \frac{1}{3!}+\cdots\right)\left(\frac{1}{1!} + \frac{1}{2!}+ \frac{1}{3!}+\cdots\right) =\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{1}{ m!\,n!}$.

The logarithm of it is also a sum of sums which somewhat resembles the series expansion of the exponential function, but there, I think, the coefficients are powers of $\frac{1}{n!}$.

Answer 1

Just one observation $$\ln p=\sum_{n=1}^{\infty}\ln \left(1+\frac{1}{n!}\right)<\sum_{n=1}^{\infty}\frac{1}{n!}=e-1$$ Since $$\ln (1+x)-x< 0$$ for all $x> 0$. So, $$p<e^{e-1}\approx 5.5749\ldots$$

Additional Observation: A tighter lower and upper bound comes as below:

$$\ln p=\sum_{n\ge 1}\ln\left(1+\frac{1}{n!}\right)=\ln 2+\sum_{n\ge 2}\sum_{k\ge 1}\frac{(-1)^{k-1}}{k(n!)^k}\\ $$ Then using the inequality $$\left(\sum_{i}a_i^k\right)\le \left(\sum_{i}a_i\right)^k$$ for $a_i\ge 0$, we get (after some calculations, which is not very difficult)$$\ln 2+e-2+\frac{1}{2}\ln (1-(e-2)^2)<\ln p<\ln 2+\frac{1}{2}\ln\left(\frac{e-1}{3-e}\right)\\ \Rightarrow 2e^{e-2}\sqrt{4e-e^2-3}<p<2\sqrt{\frac{e-1}{3-e}}\\ \Rightarrow 2.8538\ldots <p< 4.9393\ldots$$

Answer 2

Note: Steven Stadnicki made a good point about the not-so-good convergence of my suggested computationa; method. I am modifying this to try to correct this.

Expanding Samrat's observation, for any positive integer $m$,

$\begin{align} \ln p &=\sum_{n=1}^{\infty}\ln \left(1+\frac{1}{n!}\right)\\ &=\sum_{n=1}^{m}\ln \left(1+\frac{1}{n!}\right)+\sum_{n=m+1}^{\infty}\ln \left(1+\frac{1}{n!}\right)\\ &= G_m+F_m\\ \text{where}\\ F_m &=\sum_{n=m+1}^{\infty}\ln \left(1+\frac{1}{n!}\right)\\ &=\sum_{n=m+1}^{\infty}\sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k(n!)^k}\\ &=\sum_{k=1}^{\infty}\sum_{n=m+1}^{\infty} \frac{(-1)^{k-1}}{k(n!)^k}\\ &=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=m+1}^{\infty} \frac{1}{(n!)^k}\\ \end{align} $

Let $f_m(k) = \sum_{n=m+1}^{\infty} \frac{1}{(n!)^k} $, so $F_m = \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k} f_m(k) $.

$f_m(1)$ is $e$ minus the start of its series, and so is transcendental.

$f_m(2)$ is $I_0(2)$ minus the start of its series, where $I_0(x)$ is the modified Bessel function of the first kind. I do not know if $I_0(2)$ is transcendendal, but $J_0(1)$ is known to be, so I would be willing to bet that $all$ the $f_m(k)$ are transcendental.

I will now get an upper bound on $f_m(k)$ to aid in the computation of $F_m$.

$f_m(k) = \sum_{n=m+1}^{\infty} \frac{1}{(n!)^k} = \frac{1}{(m!)^k}\sum_{n=m+1}^{\infty} \left(\frac{m!}{n!}\right)^k $.

If $n > m$, $\frac{n!}{m!} =\prod_{k=1}^{n-m} (m+k) \ge (m+1)^{n-m} $, so $f_m(k) \le \frac{1}{(m!)^k}\sum_{n=m+1}^{\infty} \left(\frac{1}{(m+1)^{n-m}}\right)^k = \frac{1}{(m!)^k}\sum_{n=1}^{\infty} \frac{1}{(m+1)^{nk}} = \frac{1}{(m!)^k}\frac{(m+1)^{-k}}{1-(m+1)^{-k}} = \frac{1}{(m!)^k((m+1)^k-1)} $.

Since the $f_m(k)$ are decreasing, $|F_m| < f_m(1) < \frac{1}{m!m} $.

This means that the error in using the first $m$ terms in the product is less than this bound.

This somehow seems a more obvious result than I would like, but I will have to leave it at this since I do not see a more effective way of estimating the sum.

Possibly some acceleration method could be used to actually get more accurate estimates.