x2R
note y = sin x is an odd function (i.e., sin( x) = sin(x)) and the taylor seris of y = sin x has only odd powers.
x2R
question: is y = ln(1 + x) even, odd, or neither?
f x PN x
( + 1)! x x0
.
N
So either or . So we do not know exactly what is but atleast we know that is between and
x c x0 x0 c x
c
c
x x0
and so 2 .
cI
Remark: This is a Big Theorem by Taylor. See the book for the proof. The proof uses the Mean Value Theorem.
the sum keeps on going and going
-
.
P1(x)
=
X 1 (n)( ) f x0 !
(x
n
n=0
)n x0 .
The
for = ( ) is just the Taylor series for = ( ) at = 0.
Maclaurin series y f x
y f x x0
y PN x
N
y f x x0
So
( ) = ( )+ ( )
(3)
f x PN x RN x
that is
()⇡ () f x PN x We often think of all this as:
within an error of ( ) RN x .
(
)
⇡
X N (n)( ) f x0
(
fx
!x
n
n=0
We would LIKE TO HAVE THAT
(
)
=??
X 1
(n)( ) f x0
(
fx
!x
n
n=0
In other notation:
)n x0
)n x0
a finite sum, the sum stops at
-
N.
the sum keeps on going and going
2 ( 1 1] x,
question: is y = arctan(x) even, odd, or neither?
2 [ 1 1] x,
1
Math 142
Taylor/Maclaurin Polynomials and Series
Prof. Girardi
Fix an interval in the real line (e.g., might be ( 17 19)) and let be a point in , i.e.,
+
·
·
·
+
f
(n)
( x0
)
(
x0
!x
n
)n + x0 . . .
(open form)
P1(x)
=
f
(0)
() x0
+
f
(1)( ) x0
(
0!
1! x
)+
f
(2)( ) x0
(
x0
2! x
)2
+·
·
·+
f
(n)( x0
)
(
x0
!x
)n+ x0 . . .
n
The Taylor series can also be written in closed form, by using sigma notation, as
(N)( )
)2 + · · · + f x0 (
x0
!x
N
)N x0 ,
(open form)
(0)( ) (1)( )
(2)( )
(N)( )
( )= f pN x
x0 + f 0!
x0 ( 1! x
)+ f x0
x0 ( 2! x
)2+· · ·+ f x0
x0 (
)N
! x x0
N
a finite sum, i.e. the sum stops
The th
for = ( ) at is:
Definition 4. N -order Remainder term y f x x0
( ) d=ef ( )
()
RN x
f x PN x
where = ( ) is the th-order Taylor polynomial for = ( ) at .
I
I
,
x0
I
2 x0 I .
Next consider a function, whose domain is I,
f: I !R
and
whose
derivatives
(n) : f
I
!
R
exist
on
the
interval
I
for
n
=
1, 2, 3, . . . , N .
The th
for = ( ) at is:
-
.
n=0
if and only if
lim ( ) = 0
!1 RN x
.
N
Recall 6.
lim !1
N
() RN x
=
0
if and only if lim !1
N
| ( )| RN x
=
0.
So 7. If
lim | ( )| = 0
(4)
!1 RN x
N
then
(
)
=
X 1
(n)( ) f x0
y pN x
N
X N
()=
(
pN x
cn x
n=0
)n x0
where the constants
(n)( )
= f x0
cn
!
n
are specially chosen so that derivatives match up at , i.e. the constants ’s are chosen so that:
x0
cn
( )= ( ) pN x0 f x0
(1)( ) = (1)( )
p
N
x0
f x0
(2)( ) = (2)( )
p
N
x0
f x0
...
(closed form)
(N)( ) = (N)( )
p
N
x0
f x0 .
The constant is the th Taylor coe cient of = ( ) about . The th
(closed form)
1Here we are assuming that the derivatives = (n)( ) exist for each in the interval and for each 2 N ⌘ {1 2 3 4 5 } .
yf x
x
I
n
, , , , ,...
2
Big Questions 3. For what values of x does the power (a.k.a. Taylor) series
! x.
n
n=0
1 The
for = ( ) at is the power series:
Definition 2.
Taylor series y f x x0
P1(x)
=
f
() x0
+
f
0( )( x0 x
which can also be written as
)
+
f
00( x0
)
(
x0
2! x
)2
-
.
Formula (open form) is in open form. It can also be written in closed form, by using sigma notation, as
X N (n)( )
()= pN x
f
x0 !
( x
n
n=0
)n x0 .
So = ( ) is a polynomial of degree at most and it has the form
Definition 1. N -order Taylor polynomial y f x x0
( ) = ( ) + 0( )(
00( ) ) + f x0 (
pN x f x0 f x0 x x0
2! x
which can also be written as (recall that 0! = 1)
( ), x
i.e.,
for
f
to equal to its Taylor series.
Notice
5.
Taking
the
lim