Value of the continued fraction 2 + 2/(2 + ...)

Find the value of the infinite continued fraction $x=2+\cfrac{2}{2+\cfrac{2}{2+\cdots}}.$

$1+\sqrt3$. The tail repeats the whole expression, so $x=2+2/x$, i.e. $x^2-2x-2=0$. The roots are $1\pm\sqrt3$, and since $x>2$ we take $1+\sqrt3\approx2.732$ (the convergents do approach it).