TITLE: Infinite fraction series
QUESTION [5 upvotes]: The infinite fraction $$f(n)=n+\frac{n}{n+\frac{n}{n+...}}$$ can be simplified to $$f(n)=\frac{n+\sqrt{n^2-4n}}{2}$$ However, I wanted to know if the fraction $$1+\frac{2}{3+\frac{4}{5+...}}$$ can be simplified further to get an answer using algebra or is the answer a transcendental number? On solving this in Desmos(till 25), I got 1.54149408254.
This is an image of the Desmos page.

REPLY [3 votes]: The Mathworld page on continued fraction constants gives the exact answer as
$$\frac1{\sqrt e-1}$$
which is indeed transcendental.
The value was known to Euler himself. See page 14 of this translation.