Approximation
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
3.02 Expansions in Powers of x
BASICS
B.1) The expansion of a function
f
@
x
D
in powers of
x
Given a function f
@
x
D
, the expansion of f
@
x
D
in powers of x is
a
@
0
D
+
a
@
1
D
x
+
a
@
2
D
x
2
+
a
@
3
D
x
3
+
...
+
a
@
k
D
x
k
+
a
@
k
+
1
D
x
k
+
1
+
...
where the numbers
a
@
0
D
, a
@
1
D
, a
@
2
D
,
... , a
@
k
D
, a
@
k
+
1
D
,
...
are chosen so that for every positive integer m, the function f
@
x
D
and the polynomial
a
@
0
D
+
a
@
1
D
x
+
a
@
2
D
x
2
+
a
@
3
D
x
3
+
...
+
a
@
m
D
x
m
have order of contact m at x
=
0.
That's quite a pill to swallow, but the idea is not all that hard once you've gotten some
experience.
·
B.1.a)
Take f
@
x
D
=
1
2
+
x
.
To get the expansion of f
@
x
D
in powers of x, you could try to work out the m
th
degree
polynomial that has order of contact m with f
@
x
D
at x
=
0 as you did in the previous lesson
on splines. But
Mathematica
has a builtin instruction that will do this work automatically
for you.
Here is the second degree polynomial that has order of contact 2 with f
@
x
D
at x
=
0:
Clear
@
f, x
D
;
f
@
x
_
D
=
1
2
+
x
;
Normal
@
Series
@
f
@
x
D
,
8
x, 0, 2
<DD
1
2

x
4
+
x
2
8
Here is the third degree polynomial that has order of contact 3 with f
@
x
D
at x
=
0:
Normal
@
Series
@
f
@
x
D
,
8
x, 0, 3
<DD
1
2

x
4
+
x
2
8

x
3
16
Here is the fourth degree polynomial that has order of contact 4 with f
@
x
D
at x
=
0:
Normal
@
Series
@
f
@
x
D
,
8
x, 0, 4
<DD
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32
Here is the tenth degree polynomial that has order of contact 10 with f
@
x
D
at x
=
0:
Normal
@
Series
@
f
@
x
D
,
8
x, 0, 10
<DD
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32

x
5
64
+
x
6
128

x
7
256
+
x
8
512

x
9
1024
+
x
10
2048
Look at the pattern emerging above, and give the expansion of f
@
x
D
=
1
2
+
x
in powers of x.
·
Answer:
Look again:
Table
@
Normal
@
Series
@
f
@
x
D
,
8
x, 0, k
<DD
,
8
k, 3, 6
<D êê
TableForm
1
2

x
4
+
x
2
8

x
3
16
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32

x
5
64
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32

x
5
64
+
x
6
128
Those denominators are consecutive powers of
2, and the signs alternate.
The expansion of
f
@
x
D
=
1
êH
2
+
x
L
in powers of
x is:
1
2

x
2
2
+
x
2
2
3

x
3
2
4
+
x
4
2
5
+
...
+
H

1
L
k
x
k
2
k
+
1
+
...
This is
a
@
0
D
+
a
@
1
D
x
+
a
@
2
D
x
2
+
a
@
3
D
x
3
+
...
+
a
@
k
D
x
k
+
a
@
k
+
1
D
x
k
+
1
+
...
with
a
@
k
D
=
H

1
L
k
2
k
+
1
.
Try it out:
Clear
@
a, k
D
;
a
@
k
_
D
=
H

1
L
k
2
k
+
1
;
Table
A
Sum
A
a
@
k
D
x
k
,
8
k, 0, m
<E
,
8
m, 4, 6
<E êê
TableForm
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32

x
5
64
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32

x
5
64
+
x
6
128
Compare:
Table
@
Normal
@
Series
@
f
@
x
D
,
8
x, 0, m
<DD
,
8
m, 4, 6
<D êê
TableForm
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32

x
5
64
1
2

x
4
+
x
2
8

x
3
16
+
x
4
32

x
5
64
+
x
6
128
It checks.
B.2) The expansions every literate calculus person knows:
Æ
The expansion of
1
1

x
in powers of
x
is
1
+
x
+
x
2
+
x
3
+
x
4
+
...
+
x
k
+
...
Æ
The expansion of
e
x
in powers of
x
is
1
+
x
+
x
2
ë
2
+
x
3
ë
3
!
+
...
+
x
k
ë
k
! +
...
Æ
The expansion of
Sin
@
x
D
in powers of
x
is
x

x
3
ë
3
! +
x
5
ë
5
!
+
...
+
H

1
L
k
x
2
k
+
1
ë H
2
k
+
1
L
! +
..
.
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