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)"](http://www.oswego.edu/Documents/mathematics/EulerDivSeriescfProceedings.pdf), 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?



Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant?
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.


