Closed form for a pair of continued fractions What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ?
What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ?
It does bear some resemblance to the continued fraction for $e$, which is $2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cdots}}}$. 
Another thing I was wondering: can all transcendental numbers be expressed as infinite continued fractions containing only rational numbers? Of course for almost all transcendental numbers there does not exist any method to determine all the numerators and denominators.
 A: I don't know if either of the continued fractions can be expressed in terms of common functions and constants. However, all real numbers can be expressed as a continued fractions containing only integers.  The continued fractions terminate for rational numbers, repeat for a quadratic algebraic numbers, and neither terminate nor repeat for other reals.
Shameless plug: There are many references out there for continued fractions. I wrote a short paper that is kind of dry and covers only the basics (nothing close to the results that J. M. cites), but it goes over the results that I mentioned.
A: I know how to do these.  Here is the second question.  
First, a more natural one:
$$
1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}=
\frac{1}{\displaystyle e^{1/2}\sqrt{\frac{\pi}{2}}\;\mathrm{erfc}\left(\frac{1}{\sqrt{2}}\right)}
\approx 1.525135276\cdots
$$
So the original one is
$$
1+\cfrac{2}{1+\cfrac{3}{1+\ddots}} =
\frac{1}{\displaystyle \frac{1}{ e^{1/2}\sqrt{\frac{\pi}{2}}\;\mathrm{erfc}\left(\frac{1}{\sqrt{2}}\right)}-1}
\approx 1.9042712\cdots
$$  
[erfc is here ]
