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Inﬁnite Sequences and Series
Dr. Philippe B. Laval
Kennesaw State University
Abstract
This hand out is an introduction to sequences and series. It gives the
main deﬁnitions and presents some elementary results.
1
Sequences and Series  General Concepts (8.1
&8
.2)
1.1
Inﬁnite Sequences
1.1.1
Elementary Concepts
A sequence is simply a list of numbers, written in a deﬁnite order:
a
1
,a
2
3
,...a
n
n
+1
,...
where the elements
a
i
represent numbers. In this section, we only concentrate
on inﬁnite sequences. So, every number
a
n
in the sequence has a successor
a
n
+1
;
the sequence never stops. The index of the numbers does not have to start at
1
, though it does most of the time. If we call
n
0
the value of the starting index,
then there is a number
a
n
for every
n
≥
n
0
. Thus, we can deﬁne a function
f
such that
a
n
=
f
(
n
)
. In this class, we will concentrate on sequences of real
numbers. A more formal deﬁnition of a sequence is as follows:
Deﬁnition 1 (sequence)
A sequence of real numbers is a realvalued function
f
whose domain is a subset of the nonnegative integers, that is a set of the
form
{
n
0
,n
0
+1
}
where
n
0
is an integer such that
n
0
≥
0
.. The numbers
a
n
=
f
(
n
)
are called the terms of the sequence.
The typical notation for a sequence is
(
a
n
)
,o
r
{
a
n
}
or
{
a
n
}
∞
n
=1
where
a
n
denotes the general term of the sequence.
Remark
When a sequence is given by a function
{
f
(
n
)
}
∞
n
=
n
0
, the function
f
must be deﬁned for every
n
≥
n
0
.
A sequence can be given diﬀerent ways
1
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The elements of the sequence are given.
•
A formula to generate the terms of the sequence is given.
•
A recursive formula to generate the terms of the sequence is given.
Example 2
The examples below illustrate sequences given by listing all the el
ements,
1.
{
1
,
5
,
10
,
4
,
−
98
,
1000
,
0
,
2
,...
}
.
2.
±
1
,
1
2
,
1
3
²
. Though the elements are listed, we can also guess a formula
to generate them, what is it?
3.
{−
1
,
1
,
−
1
,
1
,
−
1
,
1
}
. Though the elements are listed, we can also guess
a formula to generate them, what is it?
Example 3
The examples below illustrate sequences given by a simple formula.
Notice that
n
does not have to start at
1
.
1.
±
1
n
²
∞
n
=1
. The elements of the sequence are:
±
1
,
1
2
,
1
3
²
. The general
term of the sequence is
a
n
=
1
n
.
2.
³
√
n
−
3
´
∞
n
=3
. The elements are
³
0
,
√
1
,
√
2
,
√
3
´
. In this case,
n
could
not start at
1
.
3.
{
(
−
1)
n
}
∞
n
=2
. The elements of the sequence are
{
1
,
−
1
,
1
,
−
1
}
.
Example 4
The examples below illustrate sequences given by a recursive for
mula.
1.
±
a
1
=1
a
n
=2
a
n
−
1
+5
. We can use this formula to generate all the terms.
But the terms have to be generated in order. For example, in order to get
a
10
,weneedtoknow
a
9
, and so on. Using the formula, we get that
a
2
a
1
=7
Having found
a
2
, we can now generate
a
3
a
3
a
2
9
Andsoon
.
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 Fall '08
 BLAKELOCK
 Squeeze Theorem, Sequences And Series

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