8.2. GEOMETRIC SERIES
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begin at n = 0, we can factor out common terms and use our established formulas.
This process is illustrated in the examples in this activity.
(a) Consider the sum
∞
k=1
(2)
1
3
k
= (2)
1
3
+ (2)
1
3
2
+ (2)
1
3
3
+ · · · .
Remove the common factor of (2)
1
3
from each term and hence find the sum
of the series.
(b) Next let a and r be real numbers with −1 < r < 1. Consider the sum
∞
k=3
ar
k = ar
3 + ar
4 + ar
5 + · · · .
Remove the common factor of ar 3 from each term and find the sum of the
series.
(c) Finally, we consider the most general case. Let a and r be real numbers with
−1 < r < 1, let n be a positive integer, and consider the sum
∞
k=n
ar
k = ar
n + ar
n+1 + ar
n+2 + · · · .
Remove the common factor of ar n from each term to find the sum of the series.
⊳
Summary
In this section, we encountered the following important ideas:
• A geometric series is an infinite sum of the form
∞
k=0
ar
k
where a and r are real numbers and r 0.
• For the geometric series
∞
k=0
ar
k , its nth partial sum is
S n =
n−1
k=0
ar
k .