## Precalculus (6th Edition) Blitzer

The repeated decimal $0.\bar{6}$ can be represented in lowest form as $\frac{2}{3}$
It can be an infinite geometric sequence as: \begin{align} & 0.\bar{6}=0.6+0.06+0.006+\ldots \\ & {{a}_{1}}=\frac{6}{10},r=\frac{1}{10} \\ & {{S}_{\infty }}=\frac{\frac{6}{10}}{1-\frac{1}{10}} \end{align} Therefore, \begin{align} & {{S}_{\infty }}=\frac{\frac{6}{10}}{\frac{9}{10}} \\ & {{S}_{\infty }}=\frac{6}{9} \\ & =\frac{2}{3} \end{align} Hence, the repeated decimal can be represented in the lowest form as $\frac{2}{3}$.