Infinite Geometric Series
Sum of an infinite Geometric
Series
The sum S of an infinite geometric
series where -1 < r < 1 is given by
a1
S=
1-r
An infinite geometric series where
r ≥ 1 does not have a sum.
Example 1. Find the sum of the
infinite geometric series if it exists.
2
4
8
+ +
+ ...
3
9
27
First find the value of r to determine
if the sum exists.
2
4
a1 =
a2 =
3
9
Example 1. Find the sum of the
infinite geometric series if it exists.
2
2
4
8
a
=
1
+ +
+ ...
3
3
9
27
4
a2 =
9
r = (4/9)/(2/3) r = (2/3)
a1
2/3
= 2
Sn =
=
1-r
1-(2/3)
Example 2. Find the sum of the
infinite geometric series if it exists.
1 - 3 + 9 - 27
a1 = 1 a2 = -3
r = -3/1 = -3
The sum does not exist.
Example 3.
You pull a pendulum back and
release it. It follows a swing pattern
of 25cm, 20cm, 16cm and so on
until it comes to a rest. What is the
total distance the pendulum swings
before it comes to rest.
Example 3. Pendulum swings 25cm,
20cm,16cm,etc.Total distance travel
The swing pattern of the pendulum
forms the infinite geometric series
25, 20, 16, ...
a1 = 25 a2 = 20 r = 20/25 = 4/5
Therefore the sum exists.
Example 3. Pendulum swings 25cm,
20cm,16cm,etc.Total distance travel
a1 = 25 a2 = 20
a1
Sn =
1-r
r = 20/25 = 4/5
25
=
1-(4/5)
= 125
The sum of an infinite geometric
series can be used to express a
repeating decimal in the form a/b.
Repeating decimals such as 0.2 and
0.47 represent 0.22222... and
0.474747... respectively.
Each expression can be written as
an infinite geometric series.
Repeating decimals such as 0.2 and
0.47 represent 0.22222... and
0.474747... respectively.
Each expression can be written as
an infinite geometric series.
0.222... = 0.2+0.02+0.002+...
0.4747... = 0.47+0.0047+...
Example 4.
Express 0.12 as a rational number
of the form a/b.
Rewrite as a geometric sum
0.12 = 0.12+0.0012+0.000012+...
a1 = 0.12 a2 = 0.0012
0.0012 r =0.01
r=
0.12
Example 4.Express 0.12 as rational
a1
Sn =
1-r
=
0.12
0.99
=
0.12
1-0.01
12
=
99
4
=
33
Example 5. Evaluate

∑ 35(-1/4)n-1
n=1
Sigma notation uses the general
form of the nth term of a geometric
series, or a1rn-1.
a1 = 35
r = -1/4
Example 5. Evaluate

∑ 35(-1/4)n-1
n=1
a1 = 35
r = -1/4
35
Sn =
1-(-1/4)
= 28