TITLE: Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$
QUESTION [14 upvotes]: How can I solve for $x$:
$$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$
Any clues?

REPLY [12 votes]: $$a_{n+1} = \frac{1}{1+a_n},\ a_0=1/x,\ \sum_{n=0}^{\infty}a_n=1$$
Unfortunately, $a_n$ converges to a non zero constant $\frac{\sqrt{5}-1}{2}$, so that the sum of all $a_n$'s is infinity.