Infinite fraction series

Question

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.

Answer 1

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.