Chapter 12.3, Problem 1CFU

a.

To determine

To graph the first ten terms of the given sequence.

Explanation of Solution

Given:

The given sequence is $a_n=\frac{n-1}{n}$

Calculation:

Substituting the values of $n=1,2,3,\mathrm{......},10$ and graphing them as follows-

  

b.

To determine

To describe what happens to the value of $n$ as $a_n$ increases.

Explanation of Solution

Given:

The given sequence is $a_n=\frac{n-1}{n}$

Calculation:

As n increases, the term $\frac{1}{n}$ of the sequence decrease and get closer and closer to zero. The net value of expression is $1$ . The terms of the modeling sequence will never actually become one; however, the terms approach one as n increases without bound.

c.

To determine

To make conjecture about the sequence as $n$ approaches infinity.

Explanation of Solution

Given:

The given sequence is $a_n=\frac{n-1}{n}$

Calculation:

As n increases, the term $\frac{1}{n}$ of the sequence decrease and get closer and closer to zero. The net value of expression is $1$ . The terms of the modeling sequence will never actually become one; however, the terms approach one as n increases without bound.

  $\underset{n\to \infty }{\lim }{a}_n=\frac{\underset{n\to \infty }{\lim }1-\underset{n\to \infty }{\lim }\frac{1}{n}}{\underset{n\to \infty }{\lim }1}=\frac{1-0}{1}=1$

d.

To determine

To evaluate $\underset{n\to \infty }{\lim }\frac{n-1}{n}$

Explanation of Solution

Given:

The given sequence is $a_n=\frac{n-1}{n}$

Calculation:

As n increases, the term $\frac{1}{n}$ of the sequence decrease and get closer and closer to zero. The net value of expression is $1$ . The terms of the modeling sequence will never actually become one; however, the terms approach one as n increases without bound.

  $\underset{n\to \infty }{\lim }{a}_n=\frac{\underset{n\to \infty }{\lim }1-\underset{n\to \infty }{\lim }\frac{1}{n}}{\underset{n\to \infty }{\lim }1}=\frac{1-0}{1}=1$