representing functions by power series. a power series is said to represent a function f with a...
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![Page 1: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/1.jpg)
Representing Functions by Power Series
![Page 2: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/2.jpg)
A power series
is said to represent a function f with a domain equal to the interval I of convergence of the series if the series converges to f(x) on that interval.
That’s if:
0n
nnxa
Ixxfxan
nn
;)(0
![Page 3: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/3.jpg)
Example
)1,1(1
1
)1,1(1
1)(
0
xifx
toconvergesseriesthisbecause
onx
xffunctionthe
representsxseriespowerThen
n
![Page 4: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/4.jpg)
Theorem
cn
cxadxxf
cxnaxf
havewercrcxThenreconvergencofrediusthehaving
cxaseriespowerthebydrepresentebexfLet
n
n
n
n
nn
n
nn
0
1
0
1
0
1
)()(.2
)()(.1
:),,(.0
)()(
![Page 5: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/5.jpg)
Examples
![Page 6: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/6.jpg)
Example(1)
?)1(
1)(
2xxg
functiontherepresentsseriespowerWhat
![Page 7: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/7.jpg)
We notice that
And we know that:
)1
1(
)1(
1)(
2
xxxg
)1,1(;1
1)(
0
xxx
xfn
n
![Page 8: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/8.jpg)
Solution
1
1
1322
0
32
0
32
)1,1(;4321)1(
1
,,
)1,1(;11
1)(
:'
)1,1(1
1)(
1
n
n
n
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n
n
n
n
n
nx
xnxxxxx
getwetermbytermatingDifferenti
xxxxxxx
xf
sThat
onx
xffunctiontherepresents
xxxxxseriespowerThe
![Page 9: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/9.jpg)
Example(2)
?)1ln()( xxg
functiontherepresentsseriespowerWhat
![Page 10: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/10.jpg)
We notice that
And we know that:
)1,1(;1
1)(
0
xxx
xfn
n
cxx
dx
x
dx
)1ln(
11
![Page 11: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/11.jpg)
Solution
1
1432
0
1
0
1
0
32
0
)1,1(;)1ln(
:
)1,1(;1432
1)1ln(
00)01ln(
)1,1(;1
)1ln(
1
1)1ln(
)1,1(;11
1)(
,
n
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n
n
n
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n
xn
xx
isstatementequivalentAn
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cc
xcn
xdxxx
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and
xxxxxxx
xf
haveWe
![Page 12: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/12.jpg)
Question
What about the convergence at the end points?
![Page 13: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/13.jpg)
1. The function ln(x-1) is not defined at x = 1
2. We can show easily that the series is convergent if x = -1 (how?)
But does it converge to ln2?
The answer to this question has to
wait till we introduce Able’s Theorem
![Page 14: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/14.jpg)
Approximating ln2
![Page 15: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/15.jpg)
3ln
:
)(,sec
69115.02ln
,
69115.06
}21
(
5
}21
(
4
}21
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}21
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2
1ln(
,
}2
1ln(,
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1
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1
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8
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6
}21
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2
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2
1ln(
1,1(;1432
)1ln(
}2
1ln(2ln
,
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165432
1432
ofionapproximatanGive
Question
ionapproximatsuchin
accuracytheerrortheestimatetoablebewillwetionnexttheIn
soand
getweseriestheoftermsfourfirsttheonlyuseweIf
accuracyofdgreeanytoedapproximatbecanThus
n
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xxxxxx
haveWe
n
n
![Page 16: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/16.jpg)
Example(3)
?arctan)( xxg
functiontherepresentsseriespowerWhat
![Page 17: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/17.jpg)
We notice that
And we know that:
)1,1(;)1()(
)(1
1
1
1)(
0
2
0
2
22
xxx
xxxf
n
nn
n
n
cxx
dx
arctan1 2
![Page 18: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/18.jpg)
)1,1(
111
1
)(1
)1(
1
1)(
2
2
2642
0
2
2
x
xx
xsatisfyingxallfor
xxxx
x
xxf
n
n
nn
![Page 19: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/19.jpg)
Solution
)1,1(;12
)1(119753
)1,1(;12
)1(arctan
000arctan
)1,1(;12
)1(arctan
)1,1(;)1(1
)(1)1(1
1
;
12119753
0
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0
12
0
22
26
0
4222
xn
xxxxxxx
xn
xx
cc
xcn
xx
xdxxx
dx
xxxxxx
haveWe
nn
n
nn
n
nn
n
nn
n
n
nn
![Page 20: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/20.jpg)
Question
What about the convergence at the end points?
![Page 21: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/21.jpg)
We can show easily that the series is convergent if x = 1or x = -1 (how?)But does it converge to arctan1 = π/4 & arctan(-1) = π/4 respectively ?
The answer to this question has to
wait until after we introduce Able’s Theorem!
![Page 22: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/22.jpg)
Approximating arctan(0.5)
![Page 23: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/23.jpg)
?)(565.26
46365.013
)2
1(
11
)2
1(
9
)2
1(
7
)2
1(
5
)2
1(
3
)2
1(
)2
1()5.0arctan(
,13
)2
1(
11
)2
1(
9
)2
1(
7
)2
1(
5
)2
1(
3
)2
1(
)2
1()5.0arctan(
,
)1,1(;12
)1(119753
arctan
,
13119753
13119753
12119753
thatatarrivewedidHowelyapproximattocorresponsThis
onlytermsseventhfirstthegConsiderin
soand
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haveWen
n
![Page 24: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/24.jpg)
everywhereabsolutelyconverges
n
x
seriesThe
Example
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:
)4(
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1lim
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:
1
![Page 25: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/25.jpg)
Showing that this series converges to e
x
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eytoleadwillthatshowwillWe
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Thus
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0
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![Page 26: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/26.jpg)
x
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en
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![Page 27: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/27.jpg)
Approximating e
7.2008.0041,0151.0500.02120
1
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,
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![Page 28: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/28.jpg)
Question
Approximate 3√e
![Page 29: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/29.jpg)
Able’s Theorem
![Page 30: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/30.jpg)
],[
).(
)(
&.2
)()(
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),(;)(
0
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rxatcontinuousisfconvergesraseriestheIf
Then
rrxxaxf
Let
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![Page 31: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/31.jpg)
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Home Quiz (2)
![Page 32: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/32.jpg)
Homework
2
4
3
2
2
4
)1()()()9(
)arctan()()8(
3arctan)()7(91
1)()6(
)23ln()()5(
)3(
1)()()4(
3)()3(
23
5)()2(
3
1)()1(
,
x
xxfxf
xf
xxfx
xf
xxf
xxfxf
x
xxf
xxf
xxf
ffortionrepresentaseriespowerafindseiesgeometricafromStarting
x
![Page 33: Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d615503460f94a42b17/html5/thumbnails/33.jpg)