## Multivariable Calculus, 7th Edition

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...
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