Module 21  Sequences and Series  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 21.1: Sequences  
In this lesson you will investigate sequences by using the TI89'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 TI89 features 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 a_{n} represents the value of the nth term of the sequence. Recall that the symbol "!" represents the
is above
The first option in this menu is 1:seq(.
Press for the factorial symbol "!," which can also be found in the Catalog. The command generates a list containing the first 10 terms in the sequence. Move the cursor into the History Area and scroll to the right to see the hidden part of the list.
21.1.1 Describe the behavior of list generated by
as n gets large.
21.1.2 Take the limit of the sequence as n approaches
by entering limit(2^n/n!, n,
). Defining Convergence Because the limit of the expression that defines the sequence exists as n approaches infinity, the sequence is said to converge to the value of the limit. The sequence defined by converges to 0 as found in Question 21.1.2. If the limit 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 A graph of the sequence illustrates convergence.
A description of the new parameters is given in the first column.
TRACE the terms of the sequence.
21.1.3 What feature of the graph indicates that the sequence converges? Creating a Table of Values A sequence that is defined in the Y= Editor can be displayed in a table.
As you scroll down the table the values of u1 get smaller and smaller, which is numerical evidence that the sequence converges to zero.
21.1.4 Does the sequence
converge? Provide graphical, numerical and analytical evidence for your conclusion. 

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