Taylor Polynomials and Power Series
Taylor Polynomials and Taylor Series.
Suppose that
f
has
N
derivatives at
a.
Then let
P
N
(
x
)
=
f
(
a
)
+
!
f
(
a
)(
x
"
a
)
+
!!
f
(
a
)
2
(
x
"
a
)
2
+
!
+
f
(
N
)
(
a
)
N
!
(
x
"
a
)
N
This is the
N
th
Taylor polynomial for f centered at a
.
If
a
=
0
, then it is known as the
N
th
Maclaurin polynomial for f.
P
N
(
x
)
=
f
(0)
+
!
f
(0)
x
+
!!
f
(0)
2
x
2
+
!
+
f
(
N
)
(0)
N
!
x
N
It is easy to see that
f
(
n
)
(
a
)
=
P
N
(
n
)
(
a
)
for
0
!
n
!
N
So, we have created a polynomial whose derivatives agree with that of
f
at
a
up to
N
.
A
Taylor series is obtained by taking the limit.
So, the
Taylor series for f centered at a
is
given by the following.
T
(
x
)
=
f
(
n
)
(
a
)
n
!
(
x
!
a
)
n
n
=
0
"
#
By this means we have defined a new function such that
T
(
n
)
(
a
)
=
f
(
n
)
(
a
)
for
n
=
0,1,2,
…
.
There are two questions that we want to answer concerning these Taylor
series.
For which values of
x
do they converge?
For which values of
x
do they represent
the original function that was used to define them?
Give a function
f
that is infinitely differentiable at a point
a
, we can define the Taylor
series quite easily.
Here are a few examples.
f
(
x
)
=
1
1
!
x
a
=
0
T
(
x
)
=
x
n
n
=
0
!
"
f
(
x
)
=
1
1
+
x
a
=
0
T
(
x
)
=
(
!
1)
n
x
n
n
=
0
"
#
f
(
x
)
=
ln(1
+
x
)
a
=
0
T
(
x
)
=
(
!
1)
n
x
n
+
1
n
+
1
n
=
0
"
#
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(
x
)
=
arctan(
x
)
a
=
0
T
(
x
)
=
(
!
1)
n
x
2
n
+
1
2
n
+
1
n
=
0
"
#
f
(
x
)
=
e
x
a
=
0
T
(
x
)
=
x
n
n
!
n
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 Spring '08
 BLOCK
 Calculus, Polynomials, Derivative, Power Series, Taylor Series, Mathematical Series, n=0

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