Series 34714.1 Sequences and Series
Main Concepts:Sequences and Series, ConvergenceA 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= 1an=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= 15 +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= 1an=S.If∑∞
n= 1anand
∑∞
n= 1bnare convergent, then∑∞
n= 1can=c∑∞
n= 1anand∑∞
n= 1(an±bn)=∑∞
n= 1an±∑∞
n= 1bn.