Calculus II-
Stewart
Dr. Berg
Spring 2010
Page 1
12.1
12.1
Sequences
The Basics
Definition
A
sequence of real numbers
is a real-valued function of natural numbers and is
denoted
a
1
,
a
2
,
…
,
a
n
,
…
{ }
,
a
n
{ }
, or,
a
n
{ }
n
=
1
∞
.
Example A
a)
a
:
N
→
R
defined by
a
(
n
)
=
n
2
.
i.e. {1, 4, 9, 16, 25, …}
b)
b
:
N
→
R
defined by
b
(
n
)
=
n
n
+
1
.
i.e. {1/2, 2/3, 3/4, 4/5, …}
Example B
Find a formula for the general term
a
n
of the sequence
1,
−
2
3
,
4
9
,
−
8
27
,
…
{ }
.
Solution
:
This sequence alternates sign. This can be accomplished with
−
1
( )
n
or
−
1
( )
n
+
1
. Note that the numerators are powers of 2 and the denominators are powers of 3.
Hence,
a
n
=
−
1
( )
n
+
1
2
3
n
−
1
.
Example C
The Fibonacci sequence
f
n
{ }
is defined recursively by
f
1
=
1
,
f
2
=
1
, and for
n
>
2
,
f
n
=
f
n
−
1
+
f
n
−
2
. Thus
f
n
{ } =
1,1,2,3,5,8,13,21,
…
{ }
.
Definition
Let
a
(
n
)
=
a
n
be a sequence of real numbers and
L
a real number. Then
lim
n
→∞
a
n
=
L
if for each
ε
>
0
there exists a number
K
∈
N
such that
n
≥
K
implies
a
n
−
L
<
. In
other words, one can make
a
n
arbitrarily close to
L
by choosing
n
to be large enough.
The point is: if a skeptic chooses some tiny distance
ε
you show that if one looks
far enough out in the sequence (past
K
) all the elements of the sequence are within that
tiny distance of the real number
L
.