Chapter 3
Sequences and Their Limits
3.0
Definition of Sequence
Informally, a
sequence
is an infinite “list” of numbers where order is important.
For example,
{
0
,
1
,
0
,
1
,
0
,
1
, . . .
}
is a sequence. Formally, we make the following
definition.
Definition
3.0.1.
A
sequence
is a function
f
:
N
→
R
.
For
n
∈
N
, we call
f
(
n
) the
n
th term
of the sequence; for the sake of intuitive notation, we will
usually denote the
n
th term of the sequence by
x
n
,
y
n
,
a
n
,
b
n
, or some other
letter with
n
as subscript, instead of
f
(
n
). We will denote the whole sequence
by
{
x
n
}
n
∈
N
or simply by
{
x
n
}
, where
x
could be any letter.
Note that a sequence does not need to have a pattern at all; however, most
of the sequence we will be working with will be defined by formulae such as
1
n
.
When we want to write down a sequence, we can specify how to compute its
n
th
term, as in
1
n
, or we can list the terms until the pattern becomes apparent,
as in
1
,
1
2
,
1
3
,
1
4
, . . .
.
We will frequently encounter sequences that are defined by
recursion
.
An
example of such a sequence is:
a
1
= 1
, a
n
+1
=
√
3 + 2
a
n
.
You wil notice that in the above example we can use what we kow of the first
term to calculate the second term. We can then determine the third term, and
so on. For recursively defined sequences calculating a particular term requires
us to already know the values of previous terms.
Diversion
.
Is there a difference between induction and recursion?
Given a sequence
{
a
1
, a
2
, a
3
, . . .
}
, a
tail
of the sequence is a sequence of the
form
{
a
k
, a
k
+1
, a
k
+2
, . . .
}
; that is, a collection of terms in the original sequence
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