Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 11 - Section 11.2 - Series - 11.2 Exercises - Page 716: 49

Answer

a) We note that the given decimal is a recurring decimal. Thus, it is probable that $x \lt 1$. b) The sum of the series is $x=1$. c) We know that the number one has two decimal representations. d) All rational numbers whose decimal representation is non-recurring have more than one decimal representation. For example, 0.4 can be written as 0.3999... and as 0.4000...

Work Step by Step

a) We note that the given decimal is a recurring decimal. Thus, it is probable that $x \lt 1$. b) Let us start by representing $x$ as the sum of a series. $x=0.9+0.09+0.009+0.0009+...$ $=9(0.1+0.01+0.001+0.0001)$ First term $a= 0.1$ and $r=\frac{1}{10}$ Sum of the series is $x=9(\frac{0.1}{1-0.1})$ $=\frac{0.9}{0.9}$ $=1$ c) We know that the number one has two decimal representations (1.000... and 0.999...). d) All rational numbers whose decimal representation is non-recurring have more than one decimal representation. For example, 0.4 can be written as 0.3999... and as 0.4000...
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