590
±
CHAPTER 10
SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS
±
10.2
SEQUENCES OF REAL NUMBERS
To this point we have considered sequences only in a peripheral manner. Here we focus
on them.
What is a sequence of real numbers?
DEFINITION 10.2.1
SEQUENCE OF REAL NUMBERS
A
sequence of real numbers
is a realvalued function deFned on the set of
positive integers.
You may Fnd this deFnition somewhat surprising, but in a moment you will see that it
makes sense.
Suppose we have a sequence of real numbers
a
1
,
a
2
,
a
3
,
...
,
a
n
,
.
What is
a
1
? It is the image of 1. What is
a
2
? It is the image of 2. What is
a
3
?Itisthe
image of 3. In general,
a
n
is the image of
n
.
By convention,
a
1
is called the
Frst term
of the sequence,
a
2
the
second term
, and
so on. More generally,
a
n
, the term with
index n
, is called the
n
th
term
.
Sequences can be deFned by giving the law of formation. ±or example:
a
n
=
1
n
is the sequence
1,
1
2
,
1
3
,
1
4
,
;
b
n
=
n
n
+
1
is the sequence
1
2
,
2
3
,
3
4
,
4
5
,
;
c
n
=
n
2
is the sequence
1, 4, 9, 16,
.
It’s like deFning
f
by giving
f
(
x
).
Sequences can be multiplied by constants; they can be added, subtracted, and
multiplied. ±rom
a
1
,
a
2
,
a
3
,
,
a
n
,
and
b
1
,
b
2
,
b
3
,
,
b
n
,
we can form
the scalar product sequence :
α
a
1
,
α
a
2
,
α
a
3
,
,
α
a
n
,
,
the sum sequence :
a
1
+
b
1
,
a
2
+
b
2
,
a
3
+
b
3
,
,
a
n
+
b
n
,
,
the difference sequence :
a
1
−
b
1
,
a
2
−
b
2
,
a
3
−
b
3
,
,
a
n
−
b
n
,
,
the product sequence :
a
1
b
1
,
a
2
b
2
,
a
3
b
3
,
,
a
n
b
n
,
.
If the
b
i
’
s
are all different from zero, we can form
the reciprocal sequence :
1
b
1
,
1
b
2
,
1
b
3
,
,
1
b
n
,
,
the quotient sequence :
a
1
b
1
,
a
2
b
2
,
a
3
b
3
,
,
a
n
b
n
,
.
The
range
of a sequence is the set of values taken on by the sequence. The range
of the sequence
a
1
,
a
2
,
a
3
,
,
a
n
,
is the set
{
a
n
:
n
=
1, 2, 3,
}
.
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View Full Document10.2
SEQUENCES OF REAL NUMBERS
±
591
Thus the range of the sequence
0, 1, 0, 1, 0, 1, 0, 1,
...
made up of alternating zeros and ones is the set
{
0, 1
}
. Can you Fnd a law of formation
a
n
for this sequence? The range of the sequence
0, 1,
−
1, 2, 2,
−
2, 3, 3, 3,
−
3, 4, 4, 4, 4,
−
4,
is the set of integers.
Boundedness and unboundedness for sequences are what they are for other real
valued functions. Thus the sequence
a
n
=
2
n
is bounded below (with greatest lower
bound 2) but unbounded above. The sequence
b
n
=
2
−
n
is bounded. It is bounded
below with greatest lower bound 0, and it is bounded above with least upper bound
1
2
.
Many of the sequences we work with have some regularity. They either have an
upward tendency or they have a downward tendency. The following terminology is
standard. The sequence
{
a
n
}
is said to be
increasing
if
a
n
<
a
n
+
1
for all
n
,
nondecreasing
if
a
n
≤
a
n
+
1
for all
n
,
decreasing
if
a
n
>
a
n
+
1
for all
n
,
nonincreasing
if
a
n
≥
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 Spring '10
 SMITH
 Real Numbers, Improper Integrals, Integrals, Order theory, Natural number

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