The sum of the first two terms is
In a similar manner, we can find additional “partial sums.”
Each term of the geometric sequence is less than the preceding one, so each ad-
ditional term is contributing less and less to the partial sum. In decimal form (to the
nearest thousandth), the first 7 terms and the 10th term are given in the table.
S 7 =
127
192
S 6 = L0.6614583.
21
32
S 5 = =0.65625,
31
48
L0.64583,
S 4 =S 3 +
1
24
=
7
12
+
1
24
=
15
24
S 3 =S 2 + =0.625,
1
12
=
1
2
+
1
12
=
7
12
L0.583,
S 2 =
1
3
+
1
6
=
1
2
=0.5.
696 CHAPTER 12 Sequences and Series
Term
Value 0.333 0.167 0.083 0.042 0.021 0.010 0.005 0.001a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 10As the table suggests, the value of a term gets closer and closer to 0 as the number of
the term increases. To express this idea, we say that as nincreases without bound
(written ), the limit of the term is 0, written
A number that can be defined as the sum of an infinite number of terms of a
geometric sequence is found by starting with the expression for the sum of a finite
number of terms.
If then as nincreases without bound, the value of gets closer and closer to 0.
As approaches 0, approaches and approaches the quotient
This limit is defined to be the sum of the infinite geometric sequence.
a 1 a 1 ra 1 r^2 a 1 r^3 Á if r< 1
a 1
1 r
,
lim
rn: 0Sn= lim
rn: 0a 111 - rn 2
1 - r
=
a 111 - 02
1 - r
=
a 1
1 - r
a 1
1 - r
.
rn 1 - rn 1 - 0 = 1, Sn
|r| 6 1, rn
Sn=
a 111 - rn 2
1 - r
lim
n:ˆan0.
n:q an
Sum of the Terms of an Infinite Geometric Sequence