460
8.2. GEOMETRIC SERIES
why S is finite and find a resulting formula for S.
⊳
From our work in Activity 8.5, we can now find the value of the geometric series
N =
12
100
+
12
100
1
100
+
12
100
1
100
2
+ · · · . In particular, using a =
12
100 and r =
1
100 , we
see that
N =
12
100
1
1 −
1
100
=
12
100
100
99
=
4
33
.
It is important to notice that a geometric sum is simply the sum of a finite number of
terms of a geometric series. In other words, the geometric sum S n for the geometric series
∞
k=0
ar
k
is
S n = a + ar + ar
2 + · · · + ar
n−1 =
n−1
k=0
ar
k .
We also call this sum S n the nth partial sum of the geometric series. We summarize our
recent work with geometric series as follows.
• A geometric series is an infinite sum of the form
a + ar + ar
2 + · · · =
∞
n=0
ar
n
,
(8.6)
where a and r are real numbers such that r 0.
• The nth partial sum S n of the geometric series is
S n = a + ar + ar
2 + · · · + ar
n−1 .
• If |r | < 1, then using the fact that S n = a
1−r n
1−r , it follows that the sum S of the
geometric series (8.6) is
S = lim
n→∞
S n = lim
n→∞
a
1 − r n
1 − r
=
a
1 − r
Activity 8.6.
The formulas we have derived for the geometric series and its partial sum so far have
assumed we begin indexing our sums at n = 0. If instead we have a sum that does not