This document discusses infinite series and their properties. It defines sequences and series, and the notation used to represent them. An infinite series is the sum of all terms in a sequence. Geometric series are introduced, which converge if the common ratio r is less than 1. The sum of a convergent geometric series is provided. Examples are given of determining if a series converges or diverges, and calculating the sum if it converges. Recurring decimals are also discussed.

1. INFINITE SERIES
CHAPTER 11

2. Convergent or Divergent
Series
Exercise 11.2
Page 541

3. 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)

4. 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)

5. WHAT SYMBOL(S) DO WE
USE
For a sequence?
{ an } represents a sequence.
For a series?
∑ an represents a series.

6. An INFINITE SERIES
(or simply a series) is an
expression of the form
a1 + a 2 + a 3 + ..... + an + ...

7. RULE
Let a ≠ 0 .
The geometric series a + ar + ar 2 + .... + ar n− 1 + .....
(i) Converges and has the sum
a
S= , if r < 1
1− r
(ii) Diverges if r ≥1

8. EXAMPLE 5 PAGE 537
Prove that the following series
converges , and find its sum:
2 2 2
2 + + 2 + .... + n−1 + ....
3 3 3

9. SOLUTION
The series converges , since it
is geometric with r < 1. Here
1
r = <1
a = 2 and 3 . The sum is
a 2 2
s= = = =3
1− r 1− 1 2
3 3

10. OR
RECURRING
DECIMALS
All nonterminating
(recurring) decimals can
be written as fractions
They are
‘rational numbers’

11. RECURRING
DECIMALS
Recurring decimals are
written by using a dot: •
0.33333 ....... = 0. 3
• •
0.35353535 ....... = 0. 3 5

12. EXERCISE 11.2 PAGE
541
Question12
Determine whether the
geometric series converges
or diverges; if it converges,
find its sum.
628
0.628 + 0.000628 + .... + n
+ ...
(1000 )

13. SOLUTION
The series converges , since it
is geometric with r < 1. Here
1
r = <1
a = 0.628 and 1000 . The
sum is
a 0.628 628
s= = =
1−r 1 999
1−
1000

14. 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.

15. SOLUTION
 5 5 
S = 10 + 2 5 + + + .......
 2 4 
 1 1 
S = 10 + 2 × 5 1 + + + .......
 2 4 
  10m 5m
 1  5
m …..
S = 10 + 10  2
1− 1 
 
 2
S = 10 + 20 = 30 Meters