I can't find anywhere via googling; is there some sort of $\sum$ like notation for infinite continued fractions? In other words, for a sum we do this:
$$ 1+x+x^2+x^3+... = \sum_{n=0}^\infty x^n $$
Easy, this notation is well known. Now, for a continued fraction:
$$ \pi= 3 + \dfrac{1}{6+\dfrac{9}{6+\dfrac{25}{6+\dfrac{49}{6+...}}}} $$
This could be defined recursively, i.e. $\pi = F_0$ where:
$$ F_n= \begin{cases} 3 + \dfrac{(2n+1)^n}{F_{n+1}} & n = 0\\ 6 + \dfrac{(2n+1)^2}{F_{n+1}} & n \geq 1 \\ \end{cases} $$
But this notation lacks the elegance of the $\sum$ example above. Even the notation for simple continued fractions isn't all that much of an improvement, i.e.
$$ \sqrt{2} = 1 + \dfrac{1}{2+\dfrac{1}{2+\dfrac{1}{2+...}}} = [1; 2, 2, 2, 2, ...] $$
Is there a better way?
Wikipedia cites a notation by Gauss: $$ x = a_0 + \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{a_3}}} $$ would be written: $$ x = a_0 + \mathop{\mathrm{K}}_{k = 1}^3 \frac{1}{a_k} $$