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