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…..