Express

Let ,x = 0.6666…

$\Rightarrow x=0.6+0.06+0.006+\ldots$

$\Rightarrow x=6(0.1+0.01+0.001+0.0001+\ldots \infty)$

$\Rightarrow x=6\left(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10000}+\ldots \infty\right)$

This is an infinite geometric series.

Here,a = 1/10 and r = 1/10

$\therefore$ Sum $=\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{\frac{1}{10}}{1-\frac{1}{10}}=\frac{1 \times 10}{9 \times 10}=\frac{1}{9}$

$\therefore x=6 \times \frac{1}{9}=\frac{6}{9}=\frac{2}{3}$

Ans: $0 . \overline{6}=\frac{2}{-}$