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Question
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
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Solution
\[0 . \overline3\]
\[\text { Let } S = 0 . \overline3\]
\[ \Rightarrow S = 0 . 3 + 0 . 03 + 0 . 003 + 0 . 0003 + 0 . 00003 + . . . \infty \]
\[ \Rightarrow S = 0 . 3\left( 1 + {10}^{- 1} + {10}^{- 2} + {10}^{- 3} + {10}^{- 4} + . . . \infty \right)\]
\[\text { S is a geometric series with the first term, a, being 1 and the common ratio, r, being } {10}^{- 1} . \]
\[ \therefore S = \frac{1}{1 - r}\]
\[ \Rightarrow S = 0 . 3\left( \frac{1}{1 - {10}^{- 1}} \right)\]
\[ \Rightarrow S = \frac{3}{9} = \frac{1}{3}\]
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