Series 347
14.1 Sequences and Series
Main Concepts:Sequences and Series, Convergence
A sequence is a function whose domain is the non-negative integers. It can be expressed as a
list of terms{an}={a 1 ,a 2 ,a 3 ,...,an,...}or by a formula that defines thenth term of the
sequence for any value ofn. A series
∑
an=
∑∞
n= 1
an=a 1 +a 2 +a 3 +···+an+···is the sum of
the terms of a sequence{an}. Associated with each series is a sequence of partial sums,{sn},
wheres 1 =a 1 ,s 2 =a 1 +a 2 ,s 3 =a 1 +a 2 +a 3 , and in general,sn=a 1 +a 2 +a 3 +···+an.
Example 1
Find the first three partial sums of the series
∑∞
n= 1
(−2)n
n^3
.
Step 1: Generate the first three terms of the sequence
{
(−2)n
n^3
}
.
a 1 =
(−2)^1
13
=−2,a 2 =
(−2)^2
23
=
4
8
=
1
2
,a 3 =
(−2)^3
33
=
− 8
27
Step 2: Find the partial sums.
s 1 =a 1 =−2,s 2 =a 1 +a 2 =− 2 +
1
2
=
− 3
2
,
s 3 =a 1 +a 2 +a 3 =− 2 +
1
2
+
− 8
27
=
− 97
54
≈− 1. 796
Example 2
Find the fifth partial sum of the series
∑∞
n= 1
5 +n^2
n+ 3
.
Step 1: Generate the first five terms of the sequence
{
5 +n^2
n+ 3
}
.
a 1 =
5 + 12
1 + 3
=
6
4
=
3
2
a 2 =
5 + 22
2 + 3
=
9
5
a 3 =
5 + 32
3 + 3
=
14
6
=
7
3
a 4 =
5 + 42
4 + 3
=
21
7
= 3 a 5 =
5 + 52
5 + 3
=
30
8
=
15
4
Step 2: The fifth partial sum isa 1 +a 2 +a 3 +a 4 +a 5 =
3
2
+
9
5
+
7
3
+ 3 +
15
4
=
743
60
.
Convergence
The series
∑
anconverges if the sequence of associated partial sums,{sn}, converges. The
limit limn→∞sn=S, whereSis a real number, and is the sum of series,
∑∞
n= 1
an=S.If
∑∞
n= 1
anand
∑∞
n= 1
bnare convergent, then
∑∞
n= 1
can=c
∑∞
n= 1
anand
∑∞
n= 1
(an±bn)=
∑∞
n= 1
an±
∑∞
n= 1
bn.