TAYLOR AND MACLAURIN how to represent certain types of functions as sums of power series You might...
38
TAYLOR AND MACLAURIN how to represent certain types of functions as sums of power series You might wonder why we would ever want to express a known function as a sum of infinitely many terms. Integration. (Easy to integrate polynomials) Finding limit Finding a sum of a series (not only geometric, telescoping) dx e x 2 2 0 1 lim x x e x x
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Transcript of TAYLOR AND MACLAURIN how to represent certain types of functions as sums of power series You might...
TAYLOR AND MACLAURIN
how to represent certain types of functions as sums of power series
You might wonder why we would ever want to express a known function as a sum of infinitely many terms.
Integration. (Easy to integrate polynomials)
Finding limit
Finding a sum of a series (not only geometric, telescoping)
dxex2
20
1lim
x
xex
x
TAYLOR AND MACLAURIN
Example: xexf )(
0n
nn
x xce 55
44
33
2210 xcxcxcxcxcc
Maclaurin series ( center is 0 )
Example:
xxf cos)(
Find Maclaurin series
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:
xxf 1tan)(
Find Maclaurin series
TAYLOR AND MACLAURIN
TERM-081
TAYLOR AND MACLAURIN
TERM-091
TAYLOR AND MACLAURIN
TERM-101
: TAYLOR AND MACLAURIN
TERM-082
)2cos(cos2
1
2
12 xx
Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-102
Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-091
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:
0 !
1
n nFind the sum of the series
TAYLOR AND MACLAURIN
TERM-102
TAYLOR AND MACLAURIN
TERM-082
TAYLOR AND MACLAURIN
Leibniz’s formula:
Example: Find the sum
0
121
12)1()(tan
n
nn
n
xx
753)(tan
7531 xxx
xx
0 12
)1(
n
n
n
7
1
5
1
3
11
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
The Binomial Series
Example:
3
3/1
!3
)23
1)(1
3
1(
3
1
81
5
DEF:
6
)3
5)(
3
2(
3
1
Example:
5
2/1
!5
)42
1)(3
2
1)(2
2
1)(1
2
1(
2
1
The Binomial Series
binomial series.
NOTE:
10
kk
kk
!11 !2
)1(
!22
kkkk
The Binomial Series
TERM-101
binomial series.
The Binomial Series
TERM-092
binomial series.
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
Example:
)1ln()( xxf
Find Maclaurin series
TAYLOR AND MACLAURIN
TERM-102
TAYLOR AND MACLAURIN
TERM-111
TAYLOR AND MACLAURIN
TERM-101
TAYLOR AND MACLAURIN
TERM-082
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Taylor series ( center is a )
TAYLOR AND MACLAURIN
TERM-091
TAYLOR AND MACLAURIN
TERM-092
TAYLOR AND MACLAURIN
TERM-082
TAYLOR AND MACLAURIN
Taylor series ( center is a )
Taylor polynomial of order n
DEF:
TAYLOR AND MACLAURIN
TERM-102
The Taylor polynomial of order 3 generated by the function f(x)=ln(3+x) at a=1 is:
Taylor polynomial of order n
DEF:
TAYLOR AND MACLAURIN
TERM-101
TAYLOR AND MACLAURIN
TERM-081
TAYLOR AND MACLAURIN
Taylor series ( center is a )
0
)(
)(!
)()(
k
kk
axk
afxf
Taylor polynomial of order n
n
k
kk
n axk
afxP
0
)(
)(!
)()(
Remainder
1
)(
)(!
)()(
nk
kk
n axk
afxR
Taylor Series )()()( xRxPxf nn
Remainder consist of infinite terms k
n
n axn
cfxR )(
)!1(
)()(
)1(
for some c between a and x.
Taylor’s Formula
REMARK: )(not )( )1()1( afcf nn Observe that :
TAYLOR AND MACLAURIN
kn
n axn
cfxR )(
)!1(
)()(
)1(
for some c between a and x.
Taylor’s Formula
kn
n xn
cfxR
)!1(
)()(
)1(
for some c between 0 and x.
Taylor’s Formula
TAYLOR AND MACLAURIN
Taylor series ( center is a )
nth-degree Taylor polynomial of f at a.
DEF:
RemainderDEF: )()()( xTxfxR nn
Example:
01
1)(
n
nxx
xf 323
03 1)( xxxxxT
n
n
654
43 )( xxxxxR
n
n
TAYLOR AND MACLAURIN
TERM-092
TAYLOR AND MACLAURIN
TERM-081
