Module 23 - Sequences and Series

Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test

Lesson 23.1: Sequences

In this lesson you will investigate sequences using the TI-83's features. You will also explore when a sequence converges by viewing the graph of a sequence and the table of associated values.

Defining a Sequence

A sequence is a function whose domain is restricted to the set of positive integers or, in some cases, the set of nonnegative integers. The numerical representation for a sequence is a list or table of values and the graphical representation is a set of discrete points. There are several features on a TI-83 that are useful in creating these representations.

Using the Sequence Command

The sequence command can be used to create a list of a finite number of terms from a sequence.

 The Sequence Command The sequence command, seq(, is found in the LIST OPS menu and in the Catalog. The syntax for this command is seq(expression,var,low,high,step) Expression is evaluated for various values of var. The first value assigned to var is low and the value of var is then repeatedly incremented by step. The last value used will be the last value of var that does not exceed high. The 5th parameter step is optional. If no step is entered, the step size defaults to one. If low = 6, high = 10, and step = 1 (the default), the TI-83 will compute the sixth term through the tenth term of the sequence given by expression and display those terms in a list.

Finding the Terms of a Sequence

Find the first ten terms of the sequence defined by , where an represents the value of the nth term of the sequence. Recall that the symbol "!" represents the
 The factorial of a whole number is the product of the series of consecutive whole numbers that begins with 1 and ends with the number: 6! = 1x2x3x4x5x6. 0! is defined to be 1.
factorial of the preceding number.

• Open the LIST OPS menu by pressing [LIST] .

The fifth option in this menu is 5:seq(, the sequence command.

• Paste this command to the Home Screen by pressing .
• Complete the command seq(2^N/N!, N, 1, 10).
• The factorial symbol, !, which is the fourth option in the PRB submenu of the MATH menu, is accessed by pressing . The factorial symbol can also be found in the Catalog.

The command generates a list containing the first 10 terms in the sequence. Move the cursor to the right by using the right arrow key to see the hidden part of the list.

• Convert the list to fractions by pressing .

 The Fraction command does not convert all decimal values to fractions. The last value in the sequence above remains a decimal after the other values are converted to fractions.

23.1.1 Describe the behavior of the sequence generated by as n gets large.

Defining Convergence

If the terms of the sequence have a limit as n approaches infinity, the sequence is said to converge to the value of the limit. The sequence defined by converges to 0 as suggested by the result in Question 23.1.1.

If the sequence grows without bound or the values jump around or oscillate and do not approach a single value, the sequence is divergent. A sequence may oscillate and converge if the oscillations become small and the values approach a single value.

Graphing Sequences

Convergence of a sequence may be illustrated by using the TI-83 Sequence Graphing mode to display the graph of the sequence.

 Sequence Graphing Mode Variables The independent variable for Sequence Graphing mode is n. The key is used to enter the variable n in Sequence Graphing mode. The expressions that define sequences in the Y= editor are u(n), v(n) and w(n).

• Select Sequence Graphing mode by pressing and highlighting Seq. Also select Dot graphing mode.

• Enter the expression 2^n/n! in the Y= editor.

• Enter the Window parameters shown below.
• Descriptions of the new parameters are given in the first column.

 Beginning value of n nMin = 1 Xmin = 0 Ymin = -1 Ending value of n nMax = 10 Xmax = 11 Ymax = 3 Where the plot starts PlotStart = 1 Xscl = 1 Yscl = 1 The increase in n from one point to the next PlotStep = 1
 The actual sequence contains infinitely many terms. To use Sequence Graphing mode we need to choose a value for nMax even though the sequence is infinite.

• Display the graph of the first 10 terms of the sequence by pressing .

TRACE the terms of the sequence by pressing .

23.1.2 What feature of the graph indicates that the sequence converges?

A Table of Values

A sequence that is defined in the Y= Editor can also be displayed in a table.

• Open the Table Setup dialog box by pressing [TBLSET] and set both TblStart and Tbl to 1.

• Press [TABLE] to create the table.
• Use the cursor movement keys to scroll down the table.

As you scroll down the table, the values of u(n) get smaller and smaller, which is numerical evidence that the sequence converges to zero. Notice that we are not forced to choose a maximum value for n to display the table.

23.1.3 Does the sequence converge? Provide graphical and numerical evidence for your conclusion.