Express

Let, x = 0.123123123….

$\Rightarrow x=0.123+0.000123+0.000000123+\ldots \infty$

$\Rightarrow x=123(0.001+0.000001+0.000000001+\ldots \infty)$

$\Rightarrow x=123\left(\frac{1}{10^{3}}+\frac{1}{10^{6}}+\frac{1}{10^{9}}+\frac{1}{10^{12}}+\ldots \infty\right)$

This is an infinite geometric series.

Here, $a=\frac{1}{10^{3}}$ and $r=\frac{1}{10^{3}}$

$\therefore \operatorname{Sum}=\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{\frac{1}{10^{3}}}{1-\frac{1}{10^{3}}}=\frac{1 \times 1000}{999 \times 1000}=\frac{1}{999}$

$\Rightarrow \mathrm{X}=123 \times \frac{1}{999}=\frac{123}{999}$

Ans : $0 . \overline{123}=\frac{123}{999}$