Computing Infinite Continued Fractions I am looking for "tricks" used to compute infinite continued fractions.
For example, $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$$ is the golden ratio since if we denote it by $x$, then we have $$x=1+\frac{1}{x},$$ which simplifies to $$x^2-x-1=0$$ 
Are there any other (different/elegant) examples of ways to compute infinite continued fractions?

This Infinite Continued Fractions can be written in many different way.
$$ \frac{1}{1} ;\frac{1}{1+\frac{1}{1}};\frac{1}{1+ \frac{1}{1+\frac{1}{1}}};\frac{1}{1+ \frac{1}{1+ \frac{1}{1+\frac{1}{1}}}} ... $$           
like this:
$$ \frac{1}{1} ;\frac{1}{2};\frac{3}{2};\frac{5}{3};\frac{8}{5};\frac{13}{8};\frac{21}{13}... $$ 
there is a Infinite series: $$\frac{13}{8} + \sum_{n=0}^{\infty} \frac{(-1)^{n+1}(2n + 1)!}{(n + 2)!n!(4)^{2n+3}}$$
the limit of this sequance is $\phi=\frac{1 + \sqrt{5}}{2} = 1.680339887...$  
also $\phi=\frac{1 + \sqrt{5}}{2}$ is a root of $\phi^{2}-\phi-1=0$
