Series 349Example 1
Determine whether the series 1+
3
2
+
9
4
+
27
8
+···converges. 1+3
2
+
9
4
+
27
8
+···is ageometric series witha=1 andr=
3
2
. Sincer>1, the series diverges.
Example 2
Find the tenth partial sum of the series 12+ 9 +
27
4
+
81
16
+···.
While it is possible to extend the terms of the series and directly compute the tenth partial
sum, it is quicker to recognize that this is a geometric series. The ratio of any two subsequent
terms isr=
3
4
and the first term isa=12.s 10 =12
(
1 −(
3
4) 10 )1 −
3
4
≈ 45. 297
Example 3
Find the sum of the series 12+ 9 +
27
4
+
81
16
+···Since 12+ 9 +27
4
+
81
16
+···is a geometricseries witha=12 andr=
3
4
,S=
a
1 −r=
12
1 −
3
4
=48.
Decimal Expansion
The rational number equal to the repeating decimal is the sum of the geometric series that
represents the repeating decimal.
Example
Find the rational number equivalent to 3. 8 76.
Step 1: 3. 876 = 3. 8 +. 076 +. 00076 +···=
38
10
+
76
103
+
76
105
+
76
107
+···=
38
10
+
76
∑∞n= 11
102 n+^1Step 2:
∑∞n= 11
102 n+^1
is a geometric series witha=1
103
andr=1
102
. The sum of the series is
a
1 −r=
1
1000
1 −
1
100
=
1
990
.
Step 3: 3. 876 =
38
10
+ 76
∑∞n= 11
102 n+^1