Chapter 9, Problem 1PS

a.

To determine

Use a graphing utility to graph the first $10$ terms of the sequence.

Answer to Problem 1PS

  

Explanation of Solution

Calculation:

Consider the following sequence

  $a_n=\frac{n+1}{n^2+1}$

To graph the sequence, set the mode to sequence and dot as shown below.

  

Enter in the sequence using the $Y$ = button

The display is shown below

  

Now press WINDOW button and set the range of the axis

The display is shown below

  

Press the GRAPH button to draw the graph of the sequence

Hence, the display is shown below

  

b.

To determine

Estimate the value of $a_n$ as $n$ approaches infinity.

Answer to Problem 1PS

  $a_n\to 0 \text{as n}\to \infty$

Explanation of Solution

Calculation:

From the graph, note that as $n$ approaches infinity, the terms of the sequence approach zero.

Hence,

  $a_n\to 0 \text{as n}\to \infty$

c.

To determine

Complete the table.

Answer to Problem 1PS

  

Explanation of Solution

Calculation:

To draw the table, set the WINDOW as shown below

  

Hence, press the TABLE button and enter the value of $n$ to complete the table as shown below

  

d.

To determine

Determine the value of $a_n$ as $n$ approaches infinity.

Answer to Problem 1PS

  $a_n\to 0$

Explanation of Solution

Calculation:

From the table in part (c), note that as $n$ increases, the value of the sequence becomes closer to zero

Hence, when $n$ approaches infinity, the terms of the sequence approach zero.

Hence,

  $a_n\to 0$