10个最常见的泰勒级数展开公式common taylor series

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
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:
( ), x
i.e.,
for
f
to equal to its Taylor series.
Notice
5.
Taking
the
lim
N
!1
of
both
sides
in
equation
(3),
we
see
that
(
)
=
X 1
(n)( ) f x0
(
)n
fx
! x x0
n
the sum keeps on going and going
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)
Commonly Used Taylor Series
series
1
1
= 1+ + 2+ 3+ 4+ x x x x ...
x
X 1
=
n
x
n=0
2
3
4
x = 1+ + x + x + x +
e
x 2! 3! 4! . . .
X 1 n
=
x
!
n
n=0
2
4
6
8
cos
= 1 x +x x +x
x
2! 4! 6! 8! . . .
X 1
2n
=
( 1)n x
(2 )!
n
n=0
3
5
7
9
sin
=
x +x x +x
x
x 3! 5! 7! 9! . . .
X 1
2n 1
X 1
2n+1
=
( 1)(n 1) x
=or ( 1)n x
(2 1)!
(2 + 1)!
n
n
n=1
n=0
2
3
4
5
ln (1 + ) =
x +x x +x
x
x2
3
4
5 ...
(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
3
Taylor’s Remainder Theorem
Version
: 1
for
a
fixed
point
x
2
I
and
a
fixed
N
2
N.
3
There exists between and so that
c
x x0
( ) d=ef ( )
()
the=orem
(N+1)( ) fc
(
)(N +1)
(5)
RN x
for = ( )
cn
n
y fx
x0 N -order Maclaurin polynomial y f x
is just the th-order Taylor polynomial for = ( ) at = 0 and so it is
N
y f x x0
(
)
=
X N (n)(0) f
n
pN x
-
.
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
Fra Baidu bibliotek
f
x0 !
( x
n
n=0
)n x0 .
So = ( ) is a polynomial of degree at most and it has the form
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?
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 .
X 1
n
X 1
n
=
(
1)(n
x
1)
=or
( 1)n+1 x
n
n
n=1
n=1
3
5
7
9
tan 1 x
=
x +x
x3
5
x +x
7
9
...
X 1
2n 1
X 1
2n+1
=
( 1)(n 1) x
=or ( 1)n x
21
2 +1
n
n
n=1
n=0
when is valid/true
note this is the geometric series. just think of x as r
(
)n
fx
! x x0 .
n
n=0
So we basically want to show that (4) holds true.
How to do this? Well, this is where Mr. Taylor comes to the rescue! 2
2According to Mr. Taylor, his Remainder Theorem (see next page) was motivated by co↵eehouse conversations about works of Newton on planetary motion and works of Halley (of Halley’s comet) on roots of polynomials.
-
.
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
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.
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.,
!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
-
.
()⇡ () f x PN x
and the question is
() fx
=?? P1(x)
where
y
=
P1
() x
is
the
Taylor
series
of
y
=
() fx
at
. x0
Well, let’s think about what needs to be for ( ) fx
=??
P1
P1(x)
=
X 1 (n)( ) f x0 !
( x
)n x0
(1)
n
n=0
converge (usually the Root or Ratio test helps us out with this question). If the power/Taylor series in formula (1)
! 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
2 ( 1 1) x,
so:
=1+1+ 1 + 1 + 1 +
e
...
2!
(17x) = P1
3!
n
(17x)
4!
=
X 1
nn
17 x
e
n=0 n!
n!
n=0
x2R
w
note y = cos x is an even function (i.e., cos( x) = + cos(x)) and the taylor seris of y = cos x has only even powers.
does indeed converge at a point , does the series converge to what we would want it to converge to, i.e., does x
() fx
=
P1(x) ?
(2)
Question (2) is going to take some thought.
(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
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:
+
·
·
·
+
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
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
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