A. Alaca
MATH 1005
Winter 2010
2
INFINITE SEQUENCES AND SERIES
A
sequence
is an ordered list having a first element but no last element:
a
1
, a
2
, a
3
, ..., a
n
, ...
a
1
is the first term,
a
2
is the second term,
a
n
is the
n
th term or general term.
Each term
a
n
of an infinite sequence has a successor
a
n
+1
.
An infinite sequence is a function whose domain is the set of positive integers
N
:
f
:
N
→
R
,
f
(
n
) =
a
n
.
Notation:
The sequence
{
a
1
, a
2
, a
3
, ..., a
n
, ...
}
is denoted by
{
a
n
}
or
{
a
n
}
∞
1
.
Examples:
•{
a
n
}
=
{
1
,
4
,
9
,
16
, ...
}
•{
a
n
}
∞
n
=1
=
{
2
n
n
+ 1
}
=
{
1
,
4
3
,
6
4
,
8
5
, ...
}
•{
1
2
n
}
∞
n
=0
=
{
1
,
1
2
,
1
4
,
1
8
, ...,
1
2
n
, ...
}
•{
√
n
−
7
}
∞
n
=7
=
{
0
,
1
,
√
2
,
√
3
,
2
, ...,
√
n
−
7
, ...
}
•
f
1
= 1
, f
2
= 1
, f
n
=
f
n

1
+
f
n

2
,
n
≥
3
The last sequence is called the Fibonacci sequence. First few terms are
{
1
,
1
,
2
,
3
,
5
,
8
,
13
,
21
, ...
}
.
A sequence can be specified in three ways:
(i)
listing the first few terms followed by
···
if the pattern is obvious,
(ii)
providing a formula for the general term
a
n
as a function of
n
,
(iii)
providing a formula for calculating the term
a
n
as a function of earlier
terms
a
1
,
a
2
, ...,
a
n

1
and specify enough of the beginning terms so the proces of
computing higher terms can begin.
But there are some sequences that do not have a simple defining equation.
Example:
If
a
n
is the digit in the
n
th decimal place of the number
e
, then
{
a
n
}
is
a sequence whose first few terms are
{
7
,
1
,
8
,
2
,
8
,
1
,
8
,
2
,
8
,
4
,
5
, ...
}
This sequence does not have an equation for the
n
th term.