|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.
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 fifth option in this menu is 5:seq(, the sequence command.
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.
23.1.1 Describe the behavior of the sequence generated by
as n gets large.
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.
Convergence of a sequence may be illustrated by using the TI-83 Sequence Graphing mode to display the graph of the sequence.
Descriptions of the new parameters are given in the first column.
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.
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.
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