Chapter 8
Sequences and Series
8.1 Sequences
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a sequence?
• What does it mean for a sequence to converge?
• What does it mean for a sequence to diverge?
Introduction
We encounter sequences every day. Your monthly rent payments, the annual interest you
earn on investments, a list of your car’s miles per gallon every time you fill up; all are
examples of sequences. Other sequences with which you may be familiar include the
Fibonacci sequence
1, 1, 2, 3, 5, 8, . . .
in which each entry is the sum of the two preceding entries and the triangular numbers
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, . . .
which are numbers that correspond to the number of vertices seen in the triangles in
Figure 8.1. Sequences of integers are of such interest to mathematicians and others that
447
Sequences and Series
8.1 Sequences
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a sequence?
• What does it mean for a sequence to converge?
• What does it mean for a sequence to diverge?
Introduction
We encounter sequences every day. Your monthly rent payments, the annual interest you
earn on investments, a list of your car’s miles per gallon every time you fill up; all are
examples of sequences. Other sequences with which you may be familiar include the
Fibonacci sequence
1, 1, 2, 3, 5, 8, . . .
in which each entry is the sum of the two preceding entries and the triangular numbers
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, . . .
which are numbers that correspond to the number of vertices seen in the triangles in
Figure 8.1. Sequences of integers are of such interest to mathematicians and others that
447