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Express the given decimal number as a quotient of two integers

Let

    \[ x = 0.4444\ldots. \]

Express x as an infinite series, find the sum, and express x as a quotient of two integers.


We have

    \begin{align*}  x = 0.4444\ldots && \implies && x &= 0 + \sum_{k=1}^{\infty} \frac{4}{10^k} \\[9pt]  &&&&&= -4 + 4 \sum_{k=0}^{\infty} \frac{1}{10^k} \\[9pt]  &&&&&= -4 + 4 \left( \frac{1}{1 - \frac{1}{10}} \right) \\[9pt]  &&&&&= \frac{4}{9}. \end{align*}

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