9-5 Geometric Series
Quick Review
A geometric series is the sum of the terms of a geometric sequence. The sum
S
n
s sub n of the first n terms of a geometric series is
S
n
=
a
1
(
1
−
r
n
)
1
−
r
,
r
≠
1
.
s sub n , equals . fraction eh sub 1 . open . 1 minus , r to the n . close , over 1 minus r end fraction . comma r not equal to 1 .
When an infinite series has a finite sum, the series converges. When the series does not converge, the series diverges.
In a geometric series, when
|
r
|
<
1
,
vertical line r vertical line less than 1 comma the series converges to
S
=
a
1
1
−
r
.
s equals . fraction eh sub 1 , over 1 minus r end fraction . . when
|
r
|
≥
1
,
vertical line r vertical line greater than or equal to 1 comma the series diverges.
Example
What is the sum of the geometric series?
5
+
10
+
20
+
40
+
80
+
160
5 plus 10 plus 20 . plus . 40 plus 80 plus 160
n
=
6
,
a
1
=
5
,
n equals 6 , comma , eh sub 1 , equals 5 comma and
r
=
10
÷
5
=
2
.
r equals 10 divides 5 equals 2 .
s
6
=
5
(
1
−
2
6
)
1
−
2
Substitute
6
for
n
,
5
for
a
1
,
and
2
for
r
.
=
315
Evaluate
.
table with 2 rows and 3 columns , row1 column 1 , s sub 6 , column 2 equals . fraction 5 . open . 1 minus , 2 to the sixth . close , over 1 minus 2 end fraction , column 3 cap substitute . 6 , for , n comma 5 , for , eh sub 1 , comma , and , 2 , for , r . , row2 column 1 , , column 2 equals 315 , column 3 cap evaluate , . , end table
The sum is 315.
Exercises
Evaluate each finite series for the specified number of terms.
-
1
+
2
+
4
+
…
;
n
=
5
1 plus 2 plus 4 plus . dot dot dot semicolon , n equals 5
-
80
−
40
+
20
−
…
;
n
=
8
80 minus , 40 plus 20 , minus dot dot dot semicolon , n equals 8
-
12
+
2
+
1
3
+
…
;
n
=
4
12 plus 2 plus , 1 third , plus dot dot dot semicolon , n equals 4
Determine whether each infinite geometric series converges or diverges. If the series converges, state the sum.
-
150
+
30
+
6
+
…
150 plus 30 plus 6 plus . dot dot dot
-
2
.
2
+
2
.
42
+
2
.
662
+
…
2 . 2 plus 2 . 42 plus 2 . 662 plus dot dot dot
-
−
10
−
20
−
40
−
…
negative 10 minus . 20 minus 40 minus . dot dot dot
-
2
3
+
4
9
+
8
27
+
…
2 thirds , plus , 4 ninths , plus , 8 twenty sevenths , plus dot dot dot
