11. 7 Infinite Geometric Series
Goal: Find the sums of infinite
geometric series
Warm-up
Find the sum of the first 6 terms of the series:
100 + 50 + 25 + …
  1 
1 r 

Sn  a1

1

r


6
 1 
1   

2 
S6  100   
1
 1


2 


n
1   
64
S6  100   
1




2


 63 
 
S6  100 64 
 1 
 2 
Find the sum of the first 12 terms of the series:
41 + 44 + 47 + …
a a 
Sn  n 1 n 
 2 
an  a1  (n  1)d
a12  41  (12  1)3
a12  41  (11)3
a12  41  33
a12  74
 41  74 
S12  12

 2 
S12  6115 
S12  690
 63  2 
S6  100  
 64  1 
 63 
S6  100 
 32 
S6  196.875
Infinite Geometric Series
Consider the series 1  1  1  1  1  1  ...  1  ...
2 4 8 16 32 64
What happens as n gets very large?
n
The fraction gets closer and closer to 0.
Therefore we can find a partial sum of the first n terms of an infinite geometric
series. This only works when -1 < r < 1 (a fraction between -1 and 1).
The Sum of an infinite Geometric Series
The sum of an infinite geometric series with first term a1 and common ratio r is
given by
a1
S
1 r
Where |r| < 1. If |r| ≥ 1, the series has no sum.
Examples
Find the sum of the infinite geometric series, if it exists.
1. a1  5, r  2
No Sum
5
2. a1  3, r 
4
No Sum
3 3 3
3

   ...
3.
4 16 64
S
S
a1
1 r
3
 1
1  
 4
S
3
5
4
4
S  3 
5
S
12
 2.4
5
Examples
Find the common ratio of the infinite geometric series with the given sum and first
term.
S  10, a1  1
S  6, a1  2
a1
1 r
S
S
10 
1
1 r
10  10 r  1
 10 r  9
9
r
10
6
a1
1 r
2
1 r
6  6r  2
 6r  4
r
4 2

6 3
Write a Repeating Decimal as a
Fraction
Write 0.22222222… as a fraction.
Step 1: Write the repeating decimal as a sum.
2(0.1) + 2(0.1)2 + 2(0.1)3 + 2(0.1)4 + …
r = 0.1
Step 2: Write the rule for sum.
S
a1
1 r
Step 3: Substitute values for a1 and r.
S
0.2
1  0.1
S
0.2
0.9
S
2
9
Write a Repeating Decimal as a
Fraction
Write 0.454545454545… as a fraction.
Step 1: Write the repeating decimal as a sum.
45(0.01) + 45(0.01)2 + 45(0.01)3 + 45(0.01)4 + …
r = 0.01
Step 2: Write the rule for sum.
S
a1
1 r
Step 3: Substitute values for a1 and r.
S
0.45
1  0.01
0.45
S
0.99
S
45 5

99 11
Assignment
Worksheet 11.7