Answer:
Please see explanation.
Step-by-step explanation:
\(0.\overline{3}\)
\(=\frac{1}{10}(3)+(\frac{1}{10})^2(3)+(\frac{1}{10})^3(3)+(\frac{1}{10})^4(3)\cdots\)
\(=3 \cdot \frac{1}{10}(1+\frac{1}{10}+(\frac{1}{10})^2+(\frac{1}{10})^3 \cdots\)
\(=3 \cdot \frac{1}{10}\sum_{i=1}^{\infty}(\frac{1}{10})^{n-1}\)
\(=3 \cdot \frac{1}{10} \cdot \frac{1}{1-\frac{1}{10}}\)
\(=\frac{3}{10} \cdot \frac{10}{10-1}\)
\(=\frac{3}{10} \cdot \frac{10}{9}\)
\(=\frac{3}{9}\)
\(=\frac{1}{3}\)