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.
