TAYLOR AND MACLAURIN how to represent certain types of functions as sums of power series You might...
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![Page 1: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/1.jpg)
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
![Page 2: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/2.jpg)
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
![Page 3: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/3.jpg)
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
![Page 4: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/4.jpg)
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:
xxf 1tan)(
Find Maclaurin series
![Page 5: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/5.jpg)
TAYLOR AND MACLAURIN
TERM-081
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TAYLOR AND MACLAURIN
TERM-091
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TAYLOR AND MACLAURIN
TERM-101
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: TAYLOR AND MACLAURIN
TERM-082
)2cos(cos2
1
2
12 xx
![Page 9: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/9.jpg)
Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-102
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Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-091
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TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:
0 !
1
n nFind the sum of the series
![Page 12: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/12.jpg)
TAYLOR AND MACLAURIN
TERM-102
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TAYLOR AND MACLAURIN
TERM-082
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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
![Page 15: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/15.jpg)
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
![Page 16: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/16.jpg)
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
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The Binomial Series
binomial series.
NOTE:
10
kk
kk
!11 !2
)1(
!22
kkkk
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The Binomial Series
TERM-101
binomial series.
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The Binomial Series
TERM-092
binomial series.
![Page 20: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/20.jpg)
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
Example:
)1ln()( xxf
Find Maclaurin series
![Page 21: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/21.jpg)
TAYLOR AND MACLAURIN
TERM-102
![Page 22: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/22.jpg)
TAYLOR AND MACLAURIN
TERM-111
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TAYLOR AND MACLAURIN
TERM-101
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TAYLOR AND MACLAURIN
TERM-082
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TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
![Page 26: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/26.jpg)
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Taylor series ( center is a )
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TAYLOR AND MACLAURIN
TERM-091
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TAYLOR AND MACLAURIN
TERM-092
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TAYLOR AND MACLAURIN
TERM-082
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TAYLOR AND MACLAURIN
Taylor series ( center is a )
Taylor polynomial of order n
DEF:
![Page 31: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/31.jpg)
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:
![Page 32: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/32.jpg)
TAYLOR AND MACLAURIN
TERM-101
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TAYLOR AND MACLAURIN
TERM-081
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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 :
![Page 35: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/35.jpg)
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
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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
![Page 37: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649f515503460f94c74642/html5/thumbnails/37.jpg)
TAYLOR AND MACLAURIN
TERM-092
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TAYLOR AND MACLAURIN
TERM-081