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.

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 ]
