How do you properly notate $\frac{1}{x+\frac{1}{x+\frac{1}{x+...}}}$?

Question

I want to write $\frac{1}{x+n}$ where $n=\frac{1}{x+n}$. My current solution is $\frac{1}{x+\frac{1}{x+\frac{1}{x+...}}}$. Is this a proper notation?

I noticed that this is also equal to $\frac{x(...)+1}{x(x(...)+1)+1}$. Is there a proper way to notate this?

Answer 1

Read up on continued fractions and see that the notation you require is $[0;x,x,x,…]$ or $[0;\overline{x}]$, which indicates that the continued fraction has a period of length $1$ (as user2661923 points out).