Home » Blog » Express the decimal x = 0.142857142857142857… as the quotient of two integers Express the decimal x = 0.142857142857142857… 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
![\[ x = 0.142857142857142857\ldots. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d4e7b88f82c95df894bd0c0e7ac09717_l3.png "Rendered by QuickLaTeX.com")
as an infinite series, find the sum, and express as a quotient of two integers.![\begin{align*} x = 0.142857142857142857\ldots && \implies && x &= \sum_{k=1}^{\infty} \frac{142857}{1000000^k} \\[9pt] &&&&&= 142857 \cdot \sum_{k=0}^{\infty} \left( \frac{1}{1000000} \right)^k - 142857 \\[9pt] &&&&&= 142857 \left( \frac{1}{1-\frac{1}{1000000}} \right) - 142857 \\[9pt] &&&&&= \frac{142857}{999999} = \frac{1}{7}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1ecd061fcb737e2e2e4f4ad0acd9ba26_l3.png "Rendered by QuickLaTeX.com")
as an infinite series, find the sum, and express as a quotient of two integers.