11.6
TAYLOR POLYNOMIALS AND TAYLOR SERIES IN
x
−
a
±
677
±
11.6
TAYLOR POLYNOMIALS AND TAYLOR SERIES IN
x
−
a
So far we have encountered series expansions only in powers of
x
. Here we generalize
to expansions in powers of
x
−
a
, where
a
is an arbitrary real number. We begin with a
more general version of Taylor’s theorem.
THEOREM 11.6.1
TAYLOR’S THEOREM
If
g
has
n
+
1 continuous derivatives on an open interval
I
that contains the
point
a
, then for each
x
∈
I
,
g
(
x
)
=
g
(
a
)
+
g
±
(
a
)(
x
−
a
)
+
g
±±
(
a
)
2
!
(
x
−
a
)
2
+···+
g
(
n
)
(
a
)
n
!
(
x
−
a
)
n
+
R
n
(
x
)
where
R
n
(
x
)
=
1
n
!
±
x
a
g
(
n
+
1)
(
t
)(
x
−
t
)
n
dt
.
The polynomial
P
n
(
x
)
=
g
(
a
)
+
g
±
(
a
)(
x
−
a
)
+
g
±±
(
a
)
2
!
(
x
−
a
)
2
+···+
g
(
n
)
(
a
)
n
!
(
x
−
a
)
n
is called the
nth Taylor polynomial for g in powers of x
−
a
. In this more general setting,
the
Lagrange formula for the remainder, R
n
(
x
), takes the form
(11.6.2)
R
n
(
x
)
=
g
(
n
+
1)
(
c
)
(
n
+
1)
!
(
x
−
a
)
n
+
1
where
c
is some number between
a
and
x
.
Now let
x
∈
I
,
x
²=
a
, and let
J
be the interval that joins
a
to
x
. Then
(11.6.3)

R
n
(
x
)
≤
²
max
t
∈
J

g
(
n
+
1)
(
t
)

³

x
−
a

n
+
1
(
n
+
1)
!
.
If
R
n
(
x
)
→
0, then we have the series representation
g
(
x
)
=
g
(
a
)
+
g
±
(
a
)(
x
−
a
)
+
g
±±
(
a
)
2
!
(
x
−
a
)
2
+···+
g
(
n
)
(
a
)
n
!
(
x
−
a
)
n
+···
,
which, in sigma notation, takes the form
(11.6.4)
g
(
x
)
=
∞
∑
k
=
0
g
(
k
)
(
a
)
k
!
(
x
−
a
)
k
.
This is known as the Taylor expansion of
g
(
x
) in powers of
x
−
a
. The series on the right
is called a
Taylor series in x
−
a
.
All this differs from what you saw before only by a translation. DeFne
f
(
x
)
=
g
(
x
+
a
).
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±
CHAPTER 11
INFINITE SERIES
Then
f
(
k
)
(
x
)
=
g
(
k
)
(
x
+
a
)
and
f
(
k
)
(0)
=
g
(
k
)
(
a
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 Spring '10
 SMITH
 Polynomials, Taylor Series

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