5128_Ch08_434-471

Chapter 8 Overview
Section 8.1 Sequences 435
In the late seventeenth century, John Bernoulli discovered a rule for calculating limits of
fractions whose numerators and denominators both approach zero. The rule is known
today as l’Hôpital’s Rule, after Guillaume François Antoine de l’Hôpital (1661–1704),
Marquis de St. Mesme, a French nobleman who wrote the first differential calculus text,
where the rule first appeared in print. We will also use l’Hôpital’s Rule to compare the
rates at which functions of x grow as x becomes large.
In Chapter 5 we saw how to evaluate definite integrals of continuous functions and
bounded functions with a finite number of discontinuities on finite closed intervals. These
ideas are extended to integrals where one or both limits of integration are infinite, and to
integrals whose integrands become unbounded on the interval of integration. Sequences
are introduced in preparation for the study of infinite series in Chapter 9.
8.1
What you’ll learn about
• Defining a Sequence
• Arithmetic and Geometric
Sequences
• Graphing a Sequence
• Limit of a Sequence
. . . and why
Sequences arise frequently in
mathematics and applied fields.
Sequences
Defining a Sequence
We have seen sequences before, such as sequences x 0 , x 1 , … , x n , … of numerical approximations
generated by Newton’s method in Chapter 4. A sequence {a n } is a list of numbers
written in an explicit order. For example, in the sequence
{a n } {a 1 , a 2 , a 3 , … , a n , …},
a 1 is the first term, a 2 is the second term, a 3 is the third term, and so forth. The numbers
a 1 , a 2 , a 3 , … , a n , … are the terms of the sequence and a n is the nth term of the sequence.
We may also think of the sequence {a 1 , a 2 , a 3 , … , a n , …} as a function with domain the
set of positive integers and range {a 1 , a 2 , a 3 , … , a n , …}.
Any real-valued function with domain a subset of the set of positive integers is considered
a sequence. If the domain is finite, then the sequence is a finite sequence. Generally
we will concentrate on infinite sequences, that is, sequences with domains that are infinite
subsets of the positive integers.
EXAMPLE 1
Defining a Sequence Explicitly
Find the first six terms and the 100th term of the sequence {a n } where
SOLUTION
Set n equal to 1, 2, 3, 4, 5, 6, and we obtain
(1) a n n 2 .
n
1
(1) a 1 1 2 1 1 1 2 , a (1) 2 2 2 2 1 1 5 , a 1 1 1 1
3 , a 4 , a 5 , a 6 .
1 0 1 7 2 6 3 7
For n 100 we find
1 1
a 100 100
2 .
1 10 001
Now try Exercise 1.
The sequence of Example 1 is defined explicitly because the formula for a n is defined
in terms of n. Another way to define a sequence is recursively by giving a formula for a n
relating it to previous terms, as shown in Example 2.