Sequences

1
Sequences

• Lesson 8.1

2
Definition

• A succession of numbers
• Listed according to a given prescription or rule
• Typically written as a1, a2, an
• Often shortened to an
• Example
• 1, 3, 5, 7, 9,
• A sequence of odd numbers

3
Finding the nth Term

• We often give an expression of the general term
• That is used to find a specific term
• What is the 5th term of the above sequence?

4
Sequence As A Function

• Define an as a function
• Domain set of nonnegative integers
• Range subset of the real numbers
• Values a1, a2, called terms of the sequence
• Nth term an called the general term
• Some sequences have limits
• Consider

5
Converging Sequences

• Note Theorem 9.2 on limits of sequences
• Limit of the sum sum of limits, etc.
• Finding limit of convergent sequence
• Use table of values
• Use graph
• Use knowledge of rational functions
• Use L'Hopital's Rule

6
Divergent Sequences

• Some sequences oscillate
• Others just grow beyond bound

7
Determining Convergence

• Manipulate algebraically
• Simplify and take the limit

conjugate expressions
8
Determining Convergence

• Consider
• Use l'Hôpital's rule to take the limit of the
  function
• Note we are relating limit of a sequence from the
  limit of a continuous function

9
Bounded, Monotonic Sequences

• Note difference between
• Increasing (decreasing) sequence
• Strictly increasing (decreasing) sequence
• Table pg 500
• Note concept of bounded sequence
• Above
• Below Bounded implies convergent
• Both

10
Assignment

• Lesson 9.1
• Page 602
• Exercises 1 93 EOO