PART II. SEQUENCES OF REAL NUMBERS
II.1.
CONVERGENCE
Definition 1.
A
sequence
is a realvalued function
f
whose domain is the set positive integers
(
N
). The numbers
f
(1)
, f
(2)
,
· · ·
are called the
terms
of the sequence.
Notation
Function notation vs subscript notation:
f
(1)
≡
s
1
, f
(2)
≡
s
2
,
· · ·
, f
(
n
)
≡
s
n
,
· · ·
.
In discussing sequences the subscript notation is much more common than functional notation. We’ll
use subscript notation throughout our treatment of analysis.
Specifying a sequence
There are several ways to specify a sequence.
1. By giving the function. For example:
(a)
s
n
=
1
n
or
{
s
n
}
=
1
n
.
This is the sequence
{
1
,
1
2
,
1
3
,
1
4
, . . . ,
1
n
, . . .
}
.
(b)
s
n
=
n

1
n
.
This is the sequence
{
0
,
1
2
,
2
3
,
3
4
, . . . ,
n

1
n
, . . .
}
.
(c)
s
n
= (

1)
n
n
2
.
This is the sequence
{
1
,
4
,

9
,
16
, ... ,
(

1)
n
n
2
, . . .
}
.
2. By giving the first few terms to establish a pattern, leaving it to you to find the function. This
is risky – it might not be easy to recognize the pattern and/or you can be misled.
(a)
{
s
n
}
=
{
0
,
1
,
0
,
1
,
0
,
1
,.. .
}
. The pattern here is obvious; can you devise the function? It’s
s
n
=
1

(

1)
n
)
2
or
s
n
=
0
,
n
odd
1
,
n
even
(b)
{
s
n
}
=
2
,
5
2
,
10
3
,
17
4
,
26
5
, . . .
,
s
n
=
n
2
+ 1
n
.
(c)
{
s
n
}
=
{
2
,
4
,
8
,
16
,
32
,. ..
}
. What is
s
6
? What is the function? While you might say
64
and
s
n
= 2
n
,
the function I have in mind gives
s
6
=
π/
6:
s
n
= 2
n
+ (
n

1)(
n

2)(
n

3)(
n

4)(
n

5)
π
720

64
120
3. By a recursion formula. For example:
(a)
s
n
+1
=
1
n
+ 1
s
n
,
s
1
= 1. The first 5 terms are
1
,
1
2
,
1
6
,
1
24
,
1
120
, . . .
. Assuming that
the pattern continues
s
n
=
1
n
!
.
(b)
s
n
+1
=
1
2
(
s
n
+ 1)
,
s
1
= 1. The first 5 terms are
{
1
,
1
,
1
,
1
,
1
, ...
}
. Assuming that the
pattern continues
s
n
= 1
for all
n
;
{
s
n
}
is a “constant” sequence.
13
Definition 2.
A sequence
{
s
n
}
converges
to the number
s
if to each
>
0
there corresponds
a positive integer
N
such that

s
n

s

<
for all
n > N.
The number
s
is called the
limit
of the sequence.
Notation
“
{
s
n
}
converges to
s
”
is denoted by
lim
n
→∞
s
n
=
s,
or by
lim
s
n
=
s,
or by
s
n
→
s.
A sequence that does not converge is said to
diverge
.
Examples
Which of the sequences given above converge and which diverge; give the limits of the
convergent sequences.
THEOREM 1.
If
s
n
→
s
and
s
n
→
t
,
then
s
=
t
. That is, the limit of a convergent sequence
is unique.
Proof:
Suppose
s
=
t
. Assume
t > s
and let
=
t

s
. Since
s
n
→
s
,
there exists a positive
integer
N
1
such that

s

s
n

<
/
2 for all
n > N
1
. Since
s
n
→
t
, there exists a positive integer
N
2
such that

t

s
n

<
/
2 for all
n > N
2
. Let
N
= max
{
N
1
, N
2
}
and choose a positive integer
k > N
. Then
t

s
=

t

s

=

t

s
k
+
s
k

s
 ≤ 
t

s
k

+

s

s
k

<
2
+
2
=
=
t

s,
a contradiction. Therefore,
s
=
t
.
THEOREM 2.
If
{
s
n
}
converges, then
{
s
n
}
is bounded.
Proof:
Suppose
s
n
→
s
. There exists a positive integer
N
such that

s

s
n

<
1 for all
n > N
.
Therefore, it follows that

s
n

=

s
n

s
+
s
 ≤ 
s
n

s

+

s

<
1 +

s

for all
n > N.
Let
M
= max
{
s
1

,

s
2

, . . .,

s
N

,
1 +

s
}
.
Then

s
n

< M
for all
n
.
Therefore
{
s
n
}
is
bounded.
THEOREM 3.
Let
{
s
n
}
and
{
a
n
}
be sequences and suppose that there is a positive number
k
and a positive integer
N
such that

s
n
 ≤
k a
n
for all
n > N.
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 Fall '08
 Staff
 Real Numbers, Integers, Limit of a sequence, Sn