I Continued fractions and nested radicals

[ajawagner]

TL;DR

Isomorphism between continued fractions and nested radicals.

There appears to be a simple isomorphism between continued fractions and nested radicals.

Does anybody know more about this?

 

[anuttarasammyak]

y=\sqrt{x+\sqrt{x+\sqrt{x+...}}}
From your observation
y^2-y-x=0
y=\frac{1}{2}(1 \pm \sqrt{1+4x})
as x>0,y>0
y=\frac{1}{2}(1+\sqrt{1+4x})
But from the first formula, y(x=0) should be zero. How can we get value of y(x=1) from it which does not show us initial value ? I am afraid this formula is not defined well enough.