Chapter 8.2, Problem 35E

a.

To determine

To find: that x < 1 or x = 1.

Answer to Problem 35E

xmust be equal to one.

Explanation of Solution

Given information: Let x =0.99999.

Calculation:

Find,is x less than or equal to 1.

x =0.99999.

x appears to be less than one however it also appears to be infinitely close to 1.

The difference between 1 and x can only be determined by the value and place of the last decimal. Since the decimal repeats continuously there is no way to discern a difference between 1 and x . Because it is continuous x must be equal to 1.

x appears to be less than 1 however it also appears to be infinitely close to 1.

The difference between x and 1 can only be determined by the value and place of its last decimal, however the decimal is continuously repeating. This suggests that there can be no discernable difference between 1 and x.

Therefore x must be equal to one.

b.

To determine

To find: the value of x , sum a geometric series.

Answer to Problem 35E

The sum of the geometric series is 1 and that is value of x.

Explanation of Solution

Given information: Let x =0.99999.

Calculation:

x =0.99999.

  $\begin{array}{c}\frac{9}{10}+\frac{9}{100}+\frac{9}{{10}^3}+\frac{9}{{10}^4}+\mathrm{....}=\frac{9}{10}(1+\frac{1}{10}+\frac{1}{{10}^2}+\frac{1}{{10}^3})\\ =\frac{9}{10}{{\displaystyle \sum _{n=0}^{\infty }(\frac{1}{10})}}^n=\frac{9}{10}(\frac{1}{1-\frac{1}{10}})=\frac{9}{9}=1\end{array}$

So, the sum of the geometric series is 1 and that is value of x.

c.

To determine

To find: how much decimal representation does the number 1 have.

Answer to Problem 35E

Two.

Explanation of Solution

Given information: Let x =0.99999.

Calculation:

Since 0.9999=1. So 0.9999… and 1.000 are the two representations of 1.

d.

To determine

To find: which numbers have more than one decimal representation.

Answer to Problem 35E

All the numbers with infinitely many 9’s at the tail.

Explanation of Solution

Given information: Let x =0.99999.

Calculation:

Some examples:

0.1234599999…=0.12346.

0.0000000120001=0.000000012000099999…

So, all the numbers with infinitely many 9’s at the tail.