8.2. GEOMETRIC SERIES
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Definition 8.2. A geometric series is an infinite sum of the form
a + ar + ar
2 + · · · =
∞
n=0
ar
n .
(8.5)
The value of r in the geometric series (8.5) is called the common ratio of the series
because the ratio of the (n + 1)st term ar n to the nth term ar n−1 is always r.
Geometric series are very common in mathematics and arise naturally in many different
situations. As a familiar example, suppose we want to write the number with repeating
decimal expansion
N = 0.121212
as a rational number. Observe that
N = 0.12 + 0.0012 + 0.000012 + · · ·
=
12
100
+
12
100
1
100
+
12
100
1
100
2
+ · · · ,
which is an infinite geometric series with a =
12
100 and r =
1
100 . In the same way that we
were able to find a shortcut formula for the value of a (finite) geometric sum, we would like
to develop a formula for the value of a (infinite) geometric series. We explore this idea in
the following activity.
Activity 8.5.
Let r 1 and a be real numbers and let
S = a + ar + ar
2 + · · · ar
n−1 + · · ·
be an infinite geometric series. For each positive integer n, let
S n = a + ar + ar
2 + · · · + ar
n−1 .
Recall that
S n = a
1 − r n
1 − r
.
(a) What should we allow n to approach in order to have S n approach S?
(b) What is the value of lim
n→∞
r
n for
• |r | > 1?
• |r | < 1?
Explain.
(c) If |r | < 1, use the formula for S n and your observations in (a) and (b) to explain