11.4 Infinite Geometric Series
Consider the following geometric series:
1
1
+
2
1
+
4
1
+
1
+
8
16
+ ...
32
Even though this series has infinitely many terms, it has a finite sum. To better illustrate
this, consider the graph of the first several terms.
1
= 0.5
2
1 1
= + = 0.75
2 4
1 1 1
= + + ≈ 0.88
2 4 8
1 1 1
1
= + + +
≈ 0.94
2 4 8 16
1 1 1
1
1
= + + +
+
≈ 0.97
2 4 8 16 32
S1 =
S2
S3
S4
S5
Notice that Sn is getting closer and closer to 1, both algebraically as well as visually, as n
continues to increase.
The sum of an infinite geometric series with first term a1 and common ratio r is given by
the formula:
S =
a1
1−r
NOTE: The absolute value of r, r , must be less than 1. If r ≥ 1 , the series has no sum.
For example, above, the sum is S =
1
2
1−
1
2
= 1 , as expected.
Ex: Find the sum of the infinite geometric series.
a) 1 −
1
1
1
+
−
+ ...
4 16 64
b) 12 + 4 +
4 4
+ + ...
3 9
Ex: Find the sum of the infinite geometric series.
∞
c)
3(0.7)n −1
∑
n =1
∞
d)
2(0.1)n −1
∑
n =1
Ex: An infinite geometric series with first term a1 = 4 has a sum of 10. What is
the common ratio of the series?
Ex: An infinite geometric series with first term a1 = 5 has a sum of
the common ratio of the series?
27
. What is
5
Writing a Repeating Decimal as a Fraction
Ex:
Ex Write 0.181818…as a fraction.
Ex: Write 0.416666…as a fraction.