## Precalculus (6th Edition) Blitzer

The repeated decimal can be represented in the lowest form as $\frac{47}{99}$.
The provided decimal can be written as: \begin{align} & 0.\overline{47}=0.474747\ldots \\ & =\frac{47}{100}+\frac{47}{10000}+\frac{47}{1000000}\ldots \\ & =0.47+0.0047+0.000047+.... \end{align} So, it can be considered a geometric sequence such that ${{a}_{1}}=\frac{47}{100},r=\frac{1}{100}$ Thus, \begin{align} & {{S}_{\infty }}=\frac{\frac{47}{100}}{1-\frac{1}{100}} \\ & =\frac{\frac{47}{100}}{\frac{99}{100}} \\ & =\frac{47}{99} \end{align}