Approximation
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
3.04 Taylor's Formula
BASICS
B.1) Taylor's formula for the expansion of
f
@
x
D
in powers of
H
x

b
L
·
B.1.a)
Look at these:
Clear
@
f, b, x
D
;
Normal
@
Series
@
f
@
x
D
,
8
x, b, 3
<DD
f
@
b
D
+
H

b
+
x
L
f
£
@
b
D
+
1
2
H

b
+
x
L
2
f
££
@
b
D
+
1
6
H

b
+
x
L
3
f
H
3
L
@
b
D
Normal
@
Series
@
f
@
x
D
,
8
x, b, 8
<DD
f
@
b
D
+
H

b
+
x
L
f
£
@
b
D
+
1
2
H

b
+
x
L
2
f
££
@
b
D
+
1
6
H

b
+
x
L
3
f
H
3
L
@
b
D
+
1
24
H

b
+
x
L
4
f
H
4
L
@
b
D
+
1
120
H

b
+
x
L
5
f
H
5
L
@
b
D
+
1
720
H

b
+
x
L
6
f
H
6
L
@
b
D
+
H

b
+
x
L
7
f
H
7
L
@
b
D
5040
+
H

b
+
x
L
8
f
H
8
L
@
b
D
40320
What's the message?
·
Answer:
Look at some more:
Clear
@
f, b, x
D
;
Normal
@
Series
@
f
@
x
D
,
8
x, b, 2
<DD
f
@
b
D
+
H

b
+
x
L
f
£
@
b
D
+
1
2
H

b
+
x
L
2
f
££
@
b
D
This has order of contact
2 with
f
@
x
D
at
x
=
b.
Normal
@
Series
@
f
@
x
D
,
8
x, b, 6
<DD
f
@
b
D
+
H

b
+
x
L
f
£
@
b
D
+
1
2
H

b
+
x
L
2
f
££
@
b
D
+
1
6
H

b
+
x
L
3
f
H
3
L
@
b
D
+
1
24
H

b
+
x
L
4
f
H
4
L
@
b
D
+
1
120
H

b
+
x
L
5
f
H
5
L
@
b
D
+
1
720
H

b
+
x
L
6
f
H
6
L
@
b
D
This has order of contact
6 with
f
@
x
D
at
x
=
b.
The denominators are factorials.
The message is that the expansion of
f
@
x
D
in powers of
H
x

b
L
is
f
@
b
D
+
f
£
@
b
D H
x

b
L
+
f
££
@
b
D
H
x

b
L
2
2
!
+
f
@
3
D
@
b
D
H
x

b
L
3
3
!
+
...
+
f
@
k
D
@
b
D
H
x

b
L
k
k
!
+
...
Most everyone calls this Taylor's formula.
This is worth memorizing.
·
B.1.b)
Check out Taylor's formula for the coefficients of the expansion of f
@
x
D
=
e
x
in powers of
H
x

1
L
.
·
Answer:
The expansion of
e
x
in powers of
H
x

1
L
starts out with:
n
=
6;
b
=
1;
Clear
@
x, f
D
;
f
@
x
D
=
E
x
;
Normal
@
Series
@
f
@
x
D
,
8
x, b, n
<DD
‰ + ‰
H

1
+
x
L
+
1
2
‰
H

1
+
x
L
2
+
1
6
‰
H

1
+
x
L
3
+
1
24
‰
H

1
+
x
L
4
+
1
120
‰
H

1
+
x
L
5
+
1
720
‰
H

1
+
x
L
6
Taylor's formula for the expansion of
f
@
x
D
in powers of
H
x

1
L
is
f
@
b
D
+
f
£
@
b
D H
x

b
L
+
f
££
@
b
D
H
x

b
L
2
2
!
+
f
@
3
D
@
b
D
H
x

b
L
3
3
!
+
...
+
f
@
k
D
@
b
D
H
x

b
L
k
k
!
+
...
So Taylor's formula gives you:
Clear
@
a, k
D
;
a
@
k
_
D
:
=
H
D
@
f
@
x
D
,
8
x, k
<D ê
.x
>
b
L ê
k
!
Sum
A
a
@
k
D H
x

b
L
k
,
8
k, 0, n
<E
‰ + ‰
H

1
+
x
L
+
1
2
‰
H

1
+
x
L
2
+
1
6
‰
H

1
+
x
L
3
+
1
24
‰
H

1
+
x
L
4
+
1
120
‰
H

1
+
x
L
5
+
1
720
‰
H

1
+
x
L
6
Looking good like a calculation should.
Do it again:
n
=
12;
Normal
@
Series
@
f
@
x
D
,
8
x, b, n
<DD
‰ + ‰
H

1
+
x
L
+
1
2
‰
H

1
+
x
L
2
+
1
6
‰
H

1
+
x
L
3
+
1
24
‰
H

1
+
x
L
4
+
1
120
‰
H

1
+
x
L
5
+
1
720
‰
H

1
+
x
L
6
+
‰
H

1
+
x
L
7
5040
+
‰
H

1
+
x
L
8
40320
+
‰
H

1
+
x
L
9
362880
+
‰
H

1
+
x
L
10
3628800
+
‰
H

1
+
x
L
11
39916800
+
‰
H

1
+
x
L
12
479001600
Sum
A
a
@
k
D H
x

1
L
k
,
8
k, 0, n
<E
‰ + ‰
H

1
+
x
L
+
1
2
‰
H

1
+
x
L
2
+
1
6
‰
H

1
+
x
L
3
+
1
24
‰
H

1
+
x
L
4
+
1
120
‰
H

1
+
x
L
5
+
1
720
‰
H

1
+
x
L
6
+
‰
H

1
+
x
L
7
5040
+
‰
H

1
+
x
L
8
40320
+
‰
H

1
+
x
L
9
362880
+
‰
H

1
+
x
L
10
3628800
+
‰
H

1
+
x
L
11
39916800
+
‰
H

1
+
x
L
12
479001600
Play with these by going back to the beginning and changing
f
@
x
D
,
b, and
n.
·
B.1.c.i)
What use is Taylor's formula?
·
Answer:
Many new practitioners of calculus want to jump on this formula and use it to slam out all
expansions. But those who have been around for a while know this is not a good idea even
if you are as fast as
Mathematica
. As a matter of fact, this formula is usually the least
efficient way to obtain an expansion of a given function. Just think of the misery involved
in calculating many derivatives and plugging in.
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 Spring '08
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 Approximation

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