Math 2224 Multivariable Calculus – Chapter 8 Infinite Sequences and Series
Sec. 8.1: Sequences
I.
Sequence
A.
Definitions
1.
Def
n
: A sequence
is a list of numbers written in a definite order
a
1
,
a
2
,
a
3
,...,
a
n
,...
where
a
1
=
1
st
term
,
a
2
=
2
nd
term
,
a
3
=
3
rd
term
,
a
n
=
n
th
term
and
n
=index of
a
n
.
2. An infinite sequence
of numbers is a function whose domain is
!
+
(the set of positive
integers). You can denote the sequence by its
n
th
term
a
n
{ }
or
a
n
{ }
n
=
1
!
.
B.
Examples
1.
1,2,3,.
..,
n
,...
n
{ }
n
=
1
!
a
n
=
n
2.
1,
!
1
2
,
1
3
,
!
1
4
,...
3.
a
n
=
cos(
!
n
)
4.
a
n
=
3
n
+
1
n
+
2
5.
a
n
=
e
n
2
n
C.
Graphical Representation of Sequences
1. There are two ways to represent sequences graphically. The first method represents
the first few terms on the real axis. The second method represents the first few
terms as points on the graph of the function defining the sequence. The function is
defined only on integer inputs and the points on the graph are
(1,
a
1
), (2,
a
2
), (3,
a
3
),.
.., (
n
,
a
n
),.
..
2.
Example - # 2 from above
a
2
a
4
a
5
a
3
a
1