Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - 11.2 Exercises: 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) Explained above 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)Explained above d)Explained above
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