Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant? Much of the theory of continued fractions has been developped by Euler in the 18th century. The little survey "Euler: continued fractions and divergent series (and Nicholas Bernoulli)", mentions towards the end the continued fraction $$f(x)=\dfrac{1}{1+\dfrac{x}{1+\dfrac{x}{1+\dfrac{2x}{1+\dfrac{2x}{1+\dfrac{3x}{1+\dots}}}}}}$$ which Euler has "derived" from the divergent power series $1-x+2x^2-6x^3+24x^4-+...$ . 
For $x=1$, the continued fraction converges to a limit $f(1)\approx 0.5963475922$. I was wondering: what is known about the values $f(x)$, in particular $f(1)$? Are they known to be transcendental for $x\in\mathbb N$? Can they be expressed by other known constants?
 A: The sequence $a_n=(-1)^n n!$ satisfies $a_{n+1}+(n+1)a_n = 0$ (with $a(0)=1$).  Thus the generating function satisfies the differential equation $x^2y'+(x+1)y=1$ (where $y(0)=1$).  The unique solution is 
$$\frac{e^{\frac{1}{x}}Ei\left(1,\frac{1}{x}\right)}{x}$$
For $x=1$, the constant is $e Ei(1,1) \approx .5963473623231940743410785$.
The reason this is justified is because the sequence is Gevrey, i.e. it does not diverge 'too fast', so associated to it is a unique (generating) function which has that sequence as coefficients (asymptotically) at 0, when approached along the real line.  The modern theory that 'this all works' is essentially due to Écalle, although Lindelof had worked out quite a bit already $100$ years before.
A: Regarding the expression of $f(1)$
in terms of other known constants:
$$f(1) = -e\left(\gamma + \sum_{n\geq 1} (-1)^n \frac1{n\cdot n!} \right)
\qquad\quad \tag{$*$} \label{id}$$
$$\qquad\qquad\qquad= -e \left(\gamma - 1 + \frac1{4} - \frac1{18} + \frac1{96} - \frac1{600} + \cdots \right)$$
where $e = \sum_{n\geq0} \frac1{n!} \approx 2.718$ and
$\gamma \approx 0.577$ is the Euler-Mascheroni constant.
This infinite sum expression
follows from the identity $f(1) = -e {\rm Ei}(-1)$
and is on the Wikipedia page for the Gompertz constant.
I recommend also taking a look at the survey Euler's constant: Euler's work and modern developments by Jeffrey Lagarias,
where it appears as (2.5.11).

Since
$f(x) ``=" \sum_{n\geq 0}  (-1)^n n! x^n$
and
$e^{-1} = \sum_{n\geq 0} (-1)^n \frac{1}{n!}$,
the  identity ($*$) can be expressed more symmetrically as
$$ - \left(\sum_{n\geq 0}  (-1)^n n! \right) \left(\sum_{m\geq 0} (-1)^m \frac{1}{m!} \right) 
\quad ``=" \quad \gamma + \sum_{n\geq 1} (-1)^n \frac1{n\cdot n!}.$$
This formula suggests an argument for evaluating $f(1)$ which avoids differential equations, if one is willing to make some convenient cancellations of divergent series after rearranging the two-index summation on the left-hand side.

For arbitrary nonzero $x$, the relation ($*$) generalizes to $f(x) = - \frac1{x} e^{1/x} {\rm Ei}(-\frac1{x})$
where ${\rm Ei}(x)$ denotes the exponential integral
$${\rm Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt. $$
Using the Taylor expansion of  ${\rm Ei}(x)$,
$$f(x) = - \frac1{x} e^{1/x} \left(\gamma + \ln x + \sum_{n\geq 1} \frac{(-1)^n}{n \cdot n!} \frac1{ x^n} \right).$$
A: Not an answer per se, but check out this nice article. (How Euler did it, by Ed Sandifer)
A: A small - surely not authoritative- bit of information: http://www.groupsrv.com/science/about477640.html 
A: A nice sequence of rational numbers approximating the constant $f(1)$ is
$$ s(n) = \sum_{k=0}^{n-1} {(n+1-k)^k \over (n+2-k)^{1+k} } $$
and
$$ A = f(1) = \lim_{n \to \infty} s(n) $$
The $s(n)$ occur here as partial sums of a series which was derived by the analysis of the triangle of "Eulerian numbers"
The first five approximants are              
  [1/2, 7/12, 43/72, 323/540, 77411/129600,...]

and in steps of $10$ the partial sums are
  s( 0):     0.500000000000
  s(10):     0.596329104980
  s(20):     0.596347376719
  s(30):     0.596347363391
  s(40):     0.596347362340
  s(50):     0.596347362311
  s(60):     0.596347362325
  s(70):     0.596347362323
  s(80):     0.596347362323

(See a discussion at this question in MSE )
A: More facts and references at https://oeis.org/A073003, not the least of which is that it appears in Ramanujan's notebooks.
