Sequences

Lesson 8.1

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

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?

sin2

nn

55 sin 5

2

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 ( ) lim ( ) ?

1nn

nf n a f n

n

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

2 1

3 4

n

n

5/2 n

Divergent Sequences

• Some sequences oscillate

• Others just grow beyond bound

sin2

nn

2 5 3

2

n n

n

Determining Convergence

• Manipulate algebraically

• Simplify and take the limit

2

2 2

2

33 3

3

n n nn n n n n n

n n n

conjugate expressions

conjugate expressions

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

2

1 n

n

e

2

0

2 2lim lim lim 01 x x xx n n

x x

e e e

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

Assignment

• Lesson 9.1

• Page 602

• Exercises 1 – 93 EOO