Home » Blog » Express the decimal x = 2.02020202… as the quotient of two integers Express the decimal x = 2.02020202… as the quotient of two integers by RoRi February 29, 2016 Let Express as an infinite series, find the sum, and express as a quotient of two integers. We have Related
![\[ x = 2.02020202\ldots. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-5a28a928249e901b234105b980874f72_l3.png "Rendered by QuickLaTeX.com")
as an infinite series, find the sum, and express as a quotient of two integers.![\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*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-32af6c48a0dc73f9bf889be7b3ddd338_l3.png "Rendered by QuickLaTeX.com")
as an infinite series, find the sum, and express as a quotient of two integers.