348 STEP 4. Review the Knowledge You Need to Score High
Example 1
Determine whether the series
∑∞
n= 1
1
5 n
converges or diverges. If it converges, find its sum.
Step 1: Find the first few partial sums.
s 1 =
1
5
= 0 .2,s 2 =
1
5
+
1
25
=
6
25
= 0. 24 s 3 =
1
5
+
1
25
+
1
125
=
31
125
= 0. 248
s 4 =
1
5
+
1
25
+
1
125
+
1
625
=
156
625
= 0. 2496
Step 2: The sequence of partial sums{ 0 .2, 0.24, 0.248, 0.2496,...}converges to 0.25,
so the series converges, and its sum
∑∞
n= 1
1
5 n
= 0 .25.
Example 2
Find the sum of the series
∑∞
n= 1
(5an− 3 bn), given that
∑∞
n= 1
an=4 and
∑∞
n= 1
bn=8.
Step 1:
∑∞
n= 1
(5an− 3 bn)=
∑∞
n= 1
5 an−
∑∞
n= 1
3 bn= 5
∑∞
n= 1
an− 3
∑∞
n= 1
bn
Step 2: 5
∑∞
n= 1
an− 3
∑∞
n= 2
bn=5(4)−3(8)= 20 − 24 =− 4
14.2 Types of Series
Main Concepts:p-Series, Harmonic Series, Geometric Series, Decimal Expansion
p-Series
Thep-series is a series of the form 1+
1
2 p
+
1
3 p
+
1
4 p
+···+
1
np
+···=
∑∞
n= 1
1
np
. The
p-series converges whenp>1, and diverges when 0<p≤1.
Harmonic Series
The harmonic series 1+
1
2
+
1
3
+
1
4
+···+
1
n
+···=
∑∞
n= 1
1
n
is ap-series withp=1. The
harmonic series diverges.
Geometric Series
A geometric series is a series of the form
∑∞
n= 1
arn−^1 wherea=/0. A geometric series converges
when|r|<1. The sum of the firstnterms of a geometric series issn=
a(1−rn)
1 −r
. The sum
of the series
∑∞
n= 1
arn−^1 =nlim→∞sn=nlim→∞
a(1−rn)
1 −r
=
a
1 −r