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Express the decimal x = 2.02020202… as the quotient of two integers

Let

    \[ x = 2.02020202\ldots. \]

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


We have

    \begin{align*}  x = 2.02020202\ldots && \implies && x &= 2 + \sum_{k=1}^{\infty} \frac{2}{100^k} \\[9pt]  &&&&&= 2 + 2 \sum_{k=0}^{\infty} \left( \frac{1}{100} \right)^k \\[9pt]  &&&&&= 2 + 2  \left( \frac{1}{1-\frac{1}{100}} \right) \\[9pt]  &&&&&= \frac{200}{99}. \end{align*}

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