c
0
+
c
1
(
x

a
) +
c
2
(
x

a
)
2
+
· · ·
+
c
n
(
x

a
)
n
should satisfy these properties:
P
n,a
(
a
)
=
f
(
a
)
,
(7.1)
P
0
n,a
(
a
)
=
f
0
(
a
)
,
P
00
n,a
(
a
)
=
f
00
(
a
)
,
P
000
n,a
(
a
)
=
f
000
(
a
)
,
.
.
.
P
(
n
)
n,a
(
a
)
=
f
(
n
)
(
a
)
.
Does such a polynomial exist?
For
n
= 1, the answer is
yes
: We have seen that
L
a
(
x
) is the desired poly
nomial.
For
n
= 2, we require P
2
,a
(
x
) :=
c
0
+
c
1
(
x

a
) +
c
2
(
x

a
)
2
to be
such that P
2
,a
(
a
) :=
c
0
=
f
(
a
), P
0
2
,a
(
a
) =
c
1
+ 2
c
2
(
a

a
) =
c
1
=
f
0
(
a
), and
P
00
2
,a
(
a
) = 2
c
2
=
f
00
(
a
)
⇒
c
2
=
f
00
(
a
)
2
. This means that
P
2
,a
(
x
) :=
f
(
a
) +
f
0
(
a
)(
x

a
) +
f
00
(
a
)
2
(
x

a
)
2
is the only candidate polynomial of degree two or less to satisfy equations listed
in (7.1). On the other hand, one can easily verify that P
2
,a
(
x
) defined above
does indeed satisfy (7.1); therefore, P
2
,a
(
x
) is the unique polynomial of degree
two or less with the properties in (7.1).
In general,
P
n,a
(
x
) :=
n
X
k
=0
f
(
k
)
(
a
)
k
!
(
x

a
)
k
is the unique polynomial of degree at most
n
such that the equalities in (7.1)
all hold. We make the following definition.
Definition
7.1.1.
Suppose that
f
:
S
→
R
where
S
⊆
R
, and that
I
⊂
S
is an
open interval containing some point
a
∈
S
. Suppose also that
f
(
x
) is
n
times
differentiable at
x
=
a
. Then the
n
th degree Taylor polynomial of
f
(
x
)
centered
at
x
=
a
is the polynomial P
n,a
:
R
→
R
defined by
P
n,a
(
x
)
=
n
X
k
=0
f
(
k
)
(
a
)
k
!
(
x

a
)
k
=
f
(
a
) +
f
0
(
a
)(
x

a
) +
f
00
(
a
)
2!
(
x

a
)
2
+
f
000
(
a
)
3!
(
x

a
)
3
+
· · ·
+
f
(
n
)
(
a
)
n
!
(
x

a
)
n
.
Note that P
1
,a
(
x
) =
L
a
(
x
) is the linear approximation for
f
(
x
) centered at
x
=
a
.
112