Express

Let, x = 0.68686868…

$\Rightarrow x=0.68+0.0068+0.000068+\ldots \infty$

$\Rightarrow x=68(0.01+0.0001+\ldots \infty)$

$\Rightarrow x=68\left(\frac{1}{10^{2}}+\frac{1}{10^{4}}+\frac{1}{10^{6}}+\frac{1}{10^{8}}+\ldots \infty\right)$

Here, $a=\frac{1}{10^{2}}$ and $r=\frac{1}{10^{2}}$

$\therefore$ Sum $=\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{\frac{1}{10^{2}}}{1-\frac{1}{10^{2}}}=\frac{1 \times 100}{99 \times 100}=\frac{1}{99}$

$\Rightarrow \mathrm{X}=\left(68 \times \frac{1}{99}\right)=\frac{68}{999}=\frac{68}{999}$

Ans: $0 . \overline{68}=\frac{68}{999}$