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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