Home » Blog » Express the decimal x = 0.123123123123… as the quotient of two integers Express the decimal x = 0.123123123123… 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
![\[ x = 0.123123123123\ldots. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e78216b18255d611f3a52f0685584747_l3.png "Rendered by QuickLaTeX.com")
as an infinite series, find the sum, and express as a quotient of two integers.![\begin{align*} x = 0.123123123123\ldots && \implies && x &= \sum_{k=1}^{\infty} \frac{123}{1000^k} \\[9pt] &&&&&= 123\sum_{k=0}^{\infty} \left( \frac{1}{1000} \right)^k - 123 \\[9pt] &&&&&= 123 \left( \frac{1}{1-\frac{1}{1000}} \right) - 123 \\[9pt] &&&&&= \frac{123}{999} = \frac{41}{333}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-420091010b85c3a7b4379e778378098d_l3.png "Rendered by QuickLaTeX.com")
as an infinite series, find the sum, and express as a quotient of two integers.