Answer
$\displaystyle \frac{123}{999}$
Work Step by Step
$0.123123123 \ldots=$
$= 0.123+0.000123+0.000000123+...$
$=\displaystyle \frac{123}{1000}+\frac{123}{1,000,000}+\frac{123}{1,000,000,000}+\cdots$
... an infinite geometric series, with
$a=\displaystyle \frac{123}{1000}$ and $r=\displaystyle \frac{1}{1000}$.
$|\displaystyle \frac{1}{1000}| < 1$, so the sum exists, $S=\displaystyle \frac{a}{1-r}$
$0.123123123 \displaystyle \ldots=\frac{\frac{123}{1000}}{1-\frac{1}{1000}}=\frac{\frac{123}{1000}}{\frac{999}{1000}}$
$=\displaystyle \frac{123}{999}$
