Approximation
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
3.03 Using Expansions
BASICS
B.1) Expansions in powers of
H
x

b
L
and approximations based on them
·
B.1.a)
When you want to study behavior near a point x
=
b other than x
=
0, you can use
expansions in powers of
H
x

b
L
.
What is the expansion of a function f
@
x
D
in powers of
H
x

b
L
?
·
Answer:
Given a function
f
@
x
D
, the expansion of
f
@
x
D
in powers of
H
x

b
L
is
a
@
0
D
+
a
@
1
D
H
x

b
L
+
a
@
2
D H
x

b
L
2
+
a
@
3
D H
x

b
L
3
+
. . .
+
a
@
k
D H
x

b
L
k
+
a
@
k
+
1
D H
x

b
L
k
+
1
+
. . .
+
. . .
where the numbers
a
@
0
D
, a
@
1
D
, a
@
2
D
, . . ., a
@
k
D
, a
@
k
+
1
D
, . . .
are set so that for every positive integer
m, the function
f
@
x
D
and the polynomial
a
@
0
D
+
a
@
1
D H
x

b
L
+
a
@
2
D H
x

b
L
2
+
a
@
3
D H
x

b
L
3
+
. . .
+
a
@
m
D H
x

b
L
m
have order of contact
m at
x
=
b.
·
B.1.b.i)
Come up with the expansion of
f
@
x
D
=
1
1
+
x
in powers of
H
x

2
L
.
·
Answer:
Look at:
b
=
2;
Clear
@
f, expan3, x
D
;
f
@
x
_
D
=
1
1
+
x
;
expan3
@
x
_
D
=
Normal
@
Series
@
f
@
x
D
,
8
x, b, 3
<DD
1
3
+
2

x
9
+
1
27
H

2
+
x
L
2

1
81
H

2
+
x
L
3
The polynomial you see above has order of contact
3 with
f
@
x
D
at
x
=
2. You can check
that:
Clear
@
k
D
;
Table
@
8
D
@
f
@
x
D
,
8
x, k
<D
, D
@
expan3
@
x
D
,
8
x, k
<D<
,
8
k, 0, 4
<D ê
.x
>
2
::
1
3
,
1
3
>
,
:

1
9
,

1
9
>
,
:
2
27
,
2
27
>
,
:

2
27
,

2
27
>
,
:
8
81
, 0
>>
Look at this:
Normal
@
Series
@
f
@
x
D
,
8
x, b, 6
<DD
1
3
+
2

x
9
+
1
27
H

2
+
x
L
2

1
81
H

2
+
x
L
3
+
1
243
H

2
+
x
L
4

1
729
H

2
+
x
L
5
+
H

2
+
x
L
6
2187
This polynomial has order of contact
6 with
f
@
x
D
at
x
=
2.
The denominators are powers of
3 and the signs alternate.
The expansion of
f
@
x
D
=
1
1
+
x
in powers of
H
x

2
L
is
1
3

x

2
3
2
+
H
x

2
L
2
3
3

H
x

2
L
3
3
4
+
. . .
+
H

1
L
k
H
x

2
L
k
3
k
+
1
+
. . .
This isn't worth memorizing.
·
B.1.b.ii)
Come up with the expansion of
f
@
x
D
=
Cos
@
x
D
in powers of
I
x

p
2
M
.
·
Answer:
Look at:
Clear
@
f, x
D
;
f
@
x
_
D
=
Cos
@
x
D
;
Normal
B
Series
B
f
@
x
D
,
:
x,
p
2
, 3
>FF
p
2

x
+
1
6
J

p
2
+
x
N
3
This polynomial has order of contact
3 with
f
@
x
D
at
x
=
p
2
.
Now look at:
Normal
B
Series
B
f
@
x
D
,
:
x,
p
2
, 12
>FF
p
2

x
+
1
6
J

p
2
+
x
N
3

1
120
J

p
2
+
x
N
5
+
I

p
2
+
x
M
7
5040

I

p
2
+
x
M
9
362880
+
I

p
2
+
x
M
11
39916800
This polynomial has order of contact
12 with
f
@
x
D
at
x
=
p
2
.
The expansion of
f
@
x
D
=
Cos
@
x
D
in powers of
I
x

p
2
M
is:

I
x

p
2
M
+
I
x

p
2
M
3
3
!

I
x

p
2
M
5
5
!
+
. . .
+
H

1
L
k
+
1
I
x

p
2
M
2 k
+
1
H
2 k
+
1
L
!
+
. . .
which just happens to be the expansion of

Sin
A
x

p
2
E
in powers of
I
x

p
2
M
:
Series
B

Sin
B
x

p
2
F
,
:
x,
p
2
, 7
>F

J
x

p
2
N
+
1
6
J
x

p
2
N
3

1
120
J
x

p
2
N
5
+
I
x

p
2
M
7
5040
+
O
B
x

p
2
F
8
Because:
Cos
@
x
D
== 
Sin
B
x

p
2
F
True
Again, none of these expansions are worth memorizing.
·
B.1.c)
When you expand in powers of
H
x

0
L
, you get the expansion in powers of x:
b
=
0;
Clear
@
f, approx, x
D
;
f
@
x
_
D
=
E
x
;
approx
@
x
_
D
=
Normal
@
Series
@
f
@
x
D
,
8
x, b, 6
<DD
1
+
x
+
x
2
2
+
x
3
6
+
x
4
24
+
x
5
120
+
x
6
720
Plot
@8
f
@
x
D
, approx
@
x
D<
,
8
x, b

1, b
+
1
<
,
PlotStyle
Ø
88
Thickness
@
0.02
D
, Blue
<
,
8
Thickness
@
0.01
D
, Red
<<
,
AxesLabel
Ø
8
"x"
,
""
<
,
PlotLabel
Ø
"Powers of
H
x

0
L
centered at x
=
0"
,
Epilog
Ø
88
Blue, PointSize
@
0.04
D
, Point
@8
b, 0
<D<
,
Text
@
"expansion point"
,
8
b, 0
<
,
8
0,

2
<D<D

1.0

0.5
0.5
1.0
x
1.0
1.5
2.0
2.5
Powers of
H
x

0
L
centered at x
=
0
Here's what happens when do the same thing except this time expand in powers of
H
x

b
L
for b
=
1:
b
=
1;
Clear
@
f, approx, x
D
;
f
@
x
_
D
=
E
x
;
approx
@
x
_
D
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 Spring '08
 Staff
 Approximation, lim, Mathematica, Complex number

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