Find the rational number (fractional) representation of the repeating decimal $LaTeX: \displaystyle 0.570570570\ldots$

The first term is $LaTeX: \displaystyle a_1=0.570=\frac{570}{1000}=\frac{57}{100}$ . The length of the repeating block is 3, so the common ratio is $LaTeX: \displaystyle \frac{1}{1000}$ . Using the formula for the sum of an infinite geometric series gives $LaTeX: \displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{570}{1-\left(\frac{1}{1000}\right)}=\frac{190}{333}$ .