By utilizing the common ratio and the first term of the sequence, we
1−r
can sum the first n terms: s = a 1−r .
n
LEARNING OBJECTIVE [ edit ]
Use to find the sum of the first n terms in a geometric sequence
KEY POINTS [ edit ]
The terms of a geometric series form a geometric progression, meaning that the ratio of
successive terms in the series isconstant.
The behavior of the terms depends on the common ratio r.
The sum of a geometric series is finite as long as the terms approach zero; as the numbers near
zero, they become insignificantly small, allowing a sum to be calculated despite the series
being infinite. The sum can be computed using the self-similarity of the series.
TERM [ edit ]
geometric series
An infinite series whose terms are in a geometric progression.
Give us feedback on this content: FULL TEXT [edit ]
Geometric series are one of the simplest examples of infinite series with finite sums, although
not all of them have this property. Historically, geometric series played an important role in
the early development of calculus, and they continue to be central in the study of convergence
of series. Geometric series are used throughout mathematics, and they have important
applications in physics, engineering, biology, economics, computer science, queueing theory,
and finance.
The terms of a geometric series form a
geometric progression, meaning that the
ratio of successive terms in the series is
constant.
The following table shows several
geometric series with different common
ratios . The behavior of the terms depends
on the common ratio r:
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Geometric Series
Geometric series with different common ratios
If r is between −1 and +1 , the terms of the series become smaller and smaller,
approaching zero in the limit, and the series converges to a sum. In the case above, where
r is one half, the series has a sum of one.
If r is greater than one or less than minus one, the terms of the series become larger and
larger in magnitude. The sum of the terms also gets larger and larger, and the series has
no sum. (The series diverges. )
If r is equal to one, all of the terms of the series are the same. The series diverges. If r is
minus one, the terms take two values alternately (e.g., 2, −2, 2, −2, 2, −2,
the terms oscillates between two values (e.g., 2, 0, 2, 0, 2, 0,
…
)
…
)
. The sum of
. This is a different type of
divergence and again the series has no sum.
The sum of a geometric series is finite as long as the terms approach zero; as the numbers
near zero, they become insignificantly small, allowing a sum to be calculated despite the
series being infinite. The sum can be computed using the self-similarity of the series.
For example:
s = 1 +
2
3
+
4
8
+
9
+. . .
27
This series has common ratio 2/3. If we multiply through by this common ratio, then the
initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on:
2
3
s =
2
3
+
4
+
9
8
16
+
27
81
+. . .
This new series is the same as the original, except that the first term is missing. Subtracting
the new series (2/3)s from the original series, s cancels every term in the original but the
first:
s
−
2
3
s = 1
, so s
= 3
A similar technique can be used to evaluate any self-similarexpression.
For r ≠ 1 , the sum of the first n terms of a geometric series is:
a + ar + ar
2
+ ar
3
+
⋯
+ ar
−1
n
=
−1
∑
n
k=0
ar
k
= a
−r
1− r
1
n
,
Where a is the first term of the series, and r is the common ratio. We can derive this formula
as follows:
Let s
= a + ar + ar
Then rs
= ar + ar
Then s − rs
= a
Then s(1 − r)
2
2
+ ar
+ ar
− ar
= a(1
3
3
n
−r
n
)
+
⋯
+ ar
4
+ ar
⋯
−1
n
+ ar
n
So s
= a
−r
1−r
1
n
Therefore, by utilizing the common ratio and the first term of the sequence, we can sum the
first n terms of a sequence.
Special case: |r|
If |r|
< 1
< 1
, then we see that as n becomes very large, r becomes very small. We express this by
n
writing that as n
→ ∞ (as approaches infinity), r
n
sum of an infinitely long geometric series where s
that as s
−r
1 − r
1
= a
Therefore, for |r|
n
< 1
→
1
a
1
−r
a
=
1
−r
, s
n
0
. If we want to find a formula for the
−r
1 − r
n
1
= a
−r
1 − r
1
= a
→
n
, we can write the infinite sum as s
→
→
1
−r
a
1
1
1
a
=
1
a
−r
.
a
−r
=
1
=
1
a
−r
−r
.
, we see