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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*}

One comment

  1. Anonymous says:

    You forgot to subtract 2 when changing the series iterator, the correct procedure is below:

    *** QuickLaTeX cannot compile formula:
    \[x = 2.0202...\]
    
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: Peer's Certificate issuer is not recognized.
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:
    http://wordpress.org/extend/plugins/core-control/
    http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37
    *** QuickLaTeX cannot compile formula:
    \[x = 2 + \sum_{k=1}^{\infty}{\frac{2}{100^k}} = 2 + 2\sum_{k=0}^{\infty}{\left(\frac{1}{100}\right)^k} - 2 = 2\left(\frac{1}{1-1/100}\right) = \frac{200}{99}\]
    
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: Peer's Certificate issuer is not recognized.
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:
    http://wordpress.org/extend/plugins/core-control/
    http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

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