# Infinite Continued Fraction Notation

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: $$x = a_0 + \mathop{\mathrm{K}}_{k = 1}^3 \frac{1}{a_k}$$
• So then $\pi = 3 + \mathop{\mathrm{K}}_{k = 0}^\infty \frac{(2k+1)^2}{6}$? Feb 25, 2014 at 14:49
• However, it is pretty confusing since $\frac{(2k+1)^2}{6}\neq\frac{(4k+2)^2}{24}$.
I believe that the notation in Hatcher Topology of Numbers is the best notation. No sooner had I seen it than I got love with the notation. The notation is $$\pi=3 +{}^{1}\mkern-10mu\nearrow\mkern-8mu{}_{\normalsize 7} +{}^{1}\mkern-10mu\nearrow\mkern-8mu{}_{\normalsize 15} +{}^{1}\mkern-10mu\nearrow\mkern-8mu{}_{\normalsize 1} +{}^{1}\mkern-10mu\nearrow\mkern-8mu{}_{\normalsize 292} +{}^{1}\mkern-10mu\nearrow\mkern-8mu{}_{\normalsize 1} +{}^{1}\mkern-10mu\nearrow\mkern-8mu{}_{\normalsize 1} +\ldots$$