Taylor’s theorem in several guises
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
24 November, 2010
(at
00:53
)
Preliminaries.
A polynomial
p
() is an
“
N

topped
polynomial
”
if
Deg(
p
)
<
N
.
So
x
2

2
x
and
x
+
√
7 are 3topped, but
x
3
+
x

5
is not. The set of
N
topped polynomials is an
N

dimensional VS.
♥
1
Taylor polynomials
The setting is an open interval
J
⊂
R
.
Use
Diff
N
=
Diff
N
(
J
→
R
) for the set of
N
times dif
ferentiable fncs. This is a superset of
C
N
=
C
N
(
J
→
R
) ;
those, whose
N
th
derivative is
continuous
.
For posint
N
, a point
Q
∈
J
and
f
∈
Diff
N

1
,
define
“
the
N
th
Taylor polynomial
of
f
, cen
tered at
Q
”
1a
:
to be the
unique
N
topped polynomial
p
() whose
zero
th
through [
N

1]
st
derivatives, at
Q
, agree
with those of
f
. I.e, these
N
equations hold:
p
(
Q
) =
f
(
Q
)
,
p
0
(
Q
) =
f
0
(
Q
)
,
p
00
(
Q
) =
f
00
(
Q
)
,
. . . ,
p
(
N

1)
(
Q
) =
f
(
N

1)
(
Q
)
.
One easily checks that
p
(
x
) must be the RhS
of (1b), below. That is,
T
N
(
x
) =
T
f
N,Q
(
x
) :=
X
k
∈
[
0
.. N
)
f
(
k
)
(
Q
)
k
!
·
[
x

Q
]
k
1b
:
is the
N
th
Taylor polynomial of
f
.
♥
1
Abbreviations: VS for
vector space
;
FTC
for
Funda
mental Theorem of Calculus
;
posint
for
positive integer
.
Define the
“
the
N
th
remainder term
of
f
”
,
written
R
f
N,Q
or just
R
N
, by
f
(
x
) =:
T
N
(
x
) +
R
N
(
x
)
.
1c
:
Properties of
T
N
.
Use
T
N,Q
[
f
] as a synonym
for
T
f
N,Q
, with the same convention for the
R
N
operator and friends. Often times either the fnc,
f
, or the centerpoint,
Q
, is implicit, and we drop
it from the notation.
For a scalar
α
and
f,g
∈
Diff
N

1
, note that
T
N
[
α
·
f
] =
α
·
T
N
[
f
]
and
T
N
[
f
+
g
] =
T
N
[
f
] +
T
N
[
g
]
.
2
:
I.e,
T
N
is a
linear operator
from
Diff
N

1
to the
VS of polynomials. Also, for
f
∈
Diff
N
,
T
N
[
f
0
] =
h
T
N
[
f
]
i
0
.
3
:
That is,
T
N
commutes with differentiation;
in
symbols
T
N
D
. Letting
I
denote the identity
operator, we have that
I
=
T
N
+
R
N
. Since
I
is
linear and commutes with
D
, so is/does
R
N
.
Courtesy
FTC
, integrating (3) gives
Z
x
Q
T
N,Q
[
f
0
](
t
)
·
d
t
=
T
N,Q
[
f
](
x
)

T
N,Q
[
f
](
Q
)
=
T
f
N,Q
(
x
)

f
(
Q
)
.
There is a corresponding statement about the re
mainder term.
Taylor’s Theorem for
R
We produce two estimates
♥
2
of the remainder
term. In the results below,
N
∈
Z
+
.
4
:
Lem.
Fix an
f
∈
Diff
N
(
J
→
R
)
. At each
x
∈
J
,
function
q
7→
T
f
N,q
(
x
)
is differentiable, with value
d
d
q
T
f
N,q
(
x
)
=
f
(
N
)
(
q
)
·
[
x

q
]
N

1
[
N

1]!
.
4
0
:
And
d
d
q
R
f
N,q
(
x
) =

d
d
q
T
f
N,q
(
x
)
.
♦
♥
2
The
TayThm1
estimate, (5a), is called
Lagrange’s
form
of the
N
th
remainder term.
Our
TayThm2
, (6a), is one of the many
Integral forms
of the remainder.
Webpage
http://www.math.ufl.edu/
∼
squash/
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Taylor’s Theorem for
R
Prof. JLF King
Proof.
WLOG
x
=7. For natnum
k
and posint
‘
,
define
A
k
(
q
)
:=
f
(
k
)
(
q
)
k
!
·
[7

q
]
k
,
and
B
‘
(
q
)
:=
f
(
‘
)
(
q
)
[
‘

1]!
·
[7

q
]
‘

1
.
Now
A
0
(
q
) =
f
(
q
)
·
1, so
d
d
q
A
0
(
q
) =
f
0
(
q
) =
B
1
(
q
).
And for
k
positive,
d
d
q
A
k
(
q
) equals
1
k
!
h
f
(
k
+1)
(
q
)
·
[7

q
]
k
+
f
(
k
)
(
q
)
·
k
·
[7

q
]
k

1
·
[
1]
i
note
===
B
k
+1
(
q
)

B
k
(
q
)
.
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 Fall '07
 JURY
 Calculus, Remainder, Taylor Series, Prof. JLF King

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