INFINITE SERIESCHAPTER 11

Convergent or Divergent Series

Exercise 11.2Page 541

IN THIS LESSON YOU WILL LEARN:

Difference between a sequence and a series

Notation of a sequence and a series

Infinite geometric series

Convergent series

Divergent series

Sum of convergent series

Nonterminating decimal (Recurring Decimal)

WHAT’S THE DIFFERENCE BETWEEN A SEQUENCE AND A SERIES?

A sequence is a list (separated by commas).

A series adds the numbers in the list together.

Example:Sequence: 1, 2, 3, 4, …, n, …Series: 1 + 2 + 3 + 4 + …+ n + …

(Note that in calculus we only examine infinite sequences and series)

WHAT SYMBOL(S) DO WE USE

For a sequence?

represents a sequence.

For a series?

represents a series.

na

na

An INFINITE SERIES

(or simply a series) is an

expression of the form

...a.....aaa n321

RULE

Let . The geometric series

(i) Converges and has the sum

(ii) Diverges if

0a .....ar....arara 1n2

1 r if , r1

aS

1 r

EXAMPLE 5 PAGE 537

Prove that the following

series converges , and find

its sum:....

32

....32

32

2 1n2

SOLUTION

The series converges , since

it is geometric with r < 1.

Here a = 2 and .

The sum is 3

322

31

1

2r1

as

131

r

NONTERMINATING OR

RECURRING DECIMALS

All nonterminating (recurring) decimals can be written as fractions

They are

‘rational numbers’

RECURRING DECIMALS

Recurring decimals are written by using a dot:

3.0.......33333.0

53.0.......35353535.0

EXERCISE 11.2 PAGE 541

Question12 Determine whether the geometric series converges or diverges; if it converges, find its sum.

...)1000(

628....000628.0628.0 n

SOLUTION

The series converges , since

it is geometric with r < 1.

Here a = 0.628 and

. The sum is

999628

1000 1

1

628.0r1

as

11000

1 r

EXERCISE 11.2 PAGE 541

Question 55

A rubber ball is dropped from a height of

10 meters. If it rebounds approximately

one-half the distance after each fall, use

a geometric series to approximate the

total distance the ball travels before

coming to rest.

SOLUTION

.......

4

5

2

55210S

21

1

11010S

Meters302010S

.......

4

1

2

115210S

10m 5 m

m2

5…..