power series. a power series in x (or centered at 0) is a series of the following form:
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![Page 1: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/1.jpg)
Power Series
![Page 2: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/2.jpg)
A power series in x (or centered at 0) is a series of the following form:
1n
nnO xaa
![Page 3: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/3.jpg)
Accepted bad Convention
When writing the power series in the form shown on the right, we follow the inaccurate convention that the expression x0
should be replaced by 1, when x = 0.
0n
nnxa
![Page 4: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/4.jpg)
A power series in x-c (or centered at c) is a series of the following
form:
1
)(n
nnO cxaa
![Page 5: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/5.jpg)
Accepted bad Convention
When writing the power series in the form shown on the right, we follow the inaccurate convention that the expression (x-c)0 should be replaced by 1, when x = c.
0
)(n
nn cxa
![Page 6: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/6.jpg)
Examples I
Geometric series are power series
![Page 7: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/7.jpg)
Example (1)A power series centered at 0 and of
interval of convergence (-1,1)
11
)1,1(
11
1
1 32
0
xOr
xOr
xx
xxxx
x
n
n
n
![Page 8: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/8.jpg)
Example (2)A power series centered at 0 and and of interval of convergence
(-5,5)
)5,5(
55
15
1
)1,1(5
15
5
5
1
1
)5()
5()
5()
5(1
)5(
5
32
0
xOr
xOr
xOr
xOr
x
x
xxxx
x
x
n
n
n
![Page 9: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/9.jpg)
Example (3)A power series centered at 2 and and of interval of convergence
(-3,7)
)7,3(
73
525
15
21
)1,1(5
2
15
27
5
7
5
1
1
)5
2()
5
2()
5
2()
5
2(1
)5
2(
52
32
0
xOr
xOr
xOr
xOr
xOr
xx
toconvergesseriesThe
x
xxxx
x
x
n
n
n
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Convergence of power series
Investigating the convergence of a power series is determining for which values of x the series converges and for which values it diverges.
![Page 11: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/11.jpg)
Every power series converges at least at one point; its center
Obviously the power series
converges for x = c.
To determine the other values of x, for which the series converges, we often use the ratio test
1
)(n
nn cxa
![Page 12: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/12.jpg)
Going back to the previous examples
15
2
)(
)(lim
:
)2(
15)(
)(lim
:
)2(
1lim
1lim
:
)1(
52
152
5
15
1
1
x
satisfyingxallforconvergesabsolutelyseriespowerThe
Example
x
satisfyingxallforconvergesabsolutelyseriespowerThe
Example
xx
x
s
s
satisfyingxallforconvergesabsolutelyseriespowerThe
Example
nx
nx
n
nx
nx
n
n
n
n
n
n
n
![Page 13: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/13.jpg)
Examples II
Convergence of other power series
![Page 14: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/14.jpg)
Example (1)
1
)2(
:
n
n
n
x
seriespowertheofeconvergenctheeInvestigat
![Page 15: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/15.jpg)
),(
121
121
lim2
1)2(1)2(
lim
:
)2(
:
21
21
1
1
x
x
xn
nx
nxnx
satisfyingxallforconvergesabsolutely
n
x
seriespowerThe
n
n
n
n
n
n
![Page 16: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/16.jpg)
Convergence at the end-points of the interval
)!(
,
)1()](2[[
)1(
1)2(
)1(
11
21
21
11
21
21
Explain
seriesgalternatinfortesteconvergencmainthe
byconverges
nnseriesThe
xat
divergeswhichseriesharmonictheis
nnseriesThe
xat
n
n
n
n
nn
n
![Page 17: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/17.jpg)
Conclusion
The series converges on the
interval [- ½ , ½ )
![Page 18: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/18.jpg)
Example (2)
1 3
)2(
:
nn
nxn
seriespowertheofeconvergenctheeInvestigat
![Page 19: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/19.jpg)
51
323
13
21
13
21lim
3
2
1
3)2(
3)2)(1(
lim
:
3
)2(
:
1
1
1
x
x
x
x
n
nx
xn
xn
satisfyingxallforconvergesabsolutely
xn
seriespowerThe
n
n
n
n
n
n
nn
n
![Page 20: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/20.jpg)
Convergence at the end-points of the interval
?)(
3
)25([
5)2(
)!(
)1(3
)21(
1)1(
11
11
whydivergeswhich
nn
seriesThe
xat
Explain
testdivergencemainthebydivergeswhich
nn
seriesThe
xat
nnn
n
n
n
nn
n
![Page 21: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/21.jpg)
Conclusion
The series converges on the
interval (- 1 , 5 )
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Example (3)
0 )!1(
)4(
:
n
n
n
x
seriespowertheofeconvergenctheeInvestigat
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RinxallforconvergesabsolutelyseriestheThus
RxholdsThisn
x
nx
nx
havewe
n
x
seriespowertheGiven
n
n
n
n
n
n
,
102
1lim4
)!1()4()!2(
)4(
lim
:,
)!1(
)4(
:
1
0
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Example (4)
n
n
xn
n
seriespowertheofeconvergenctheeInvestigat
1
!
:
![Page 25: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/25.jpg)
0,
,,
0
lim1
)1(lim
!1)!1(
lim
:,
!
:
1
0
xatnamaly
RinxallforcenteritsatonlyconvergesseriestheThus
xif
nxn
nnx
xnn
xnn
havewe
xn
n
seriespowertheGiven
nn
n
n
n
n
n
![Page 26: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/26.jpg)
Theorem
A power series of the form
Is either absolutely convergent everywhere, only at its center or on some interval about its center.
0
)(n
nn cxa
![Page 27: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/27.jpg)
The three casesCase (1): we say that the series is absolutely convergent on R
or on ( -∞ , ∞) and that the radius of convergence is ∞
Case (2): we say that the series is convergent at x = c and divergent everywhere else, and that the radius of convergence is 0.
Case (3): The series is absolutely convergent at an interval of the form ( c-r,c+r), for some positive number r, and divergent on (-∞,c-r)U(c+r, ∞). In this case we say that the interval of convergence is equal to ( c-r,c+r) and the radius of convergence is equal to r. We investigate separately the convergence of the series at each of the end points of the interval
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HomeworkDetermine the interval & radius of convergence of
the given series
01
0
022
2
0
0
5
)2()5(
1
)5()4(
)!(2
)1()3(
)52()2(
!)1(
nn
n
n
nn
nn
nn
n
n
n
n
xn
n
x
n
x
n
x
xn
![Page 29: Power Series. A power series in x (or centered at 0) is a series of the following form:](https://reader036.vdocument.in/reader036/viewer/2022062516/56649d5d5503460f94a3c94f/html5/thumbnails/29.jpg)
Hints
)2,8()5(
],()4(
)3(
)6,4[)2(
)1(
51
51
onConverges
onConverges
eveywhereabsolutelyConverges
onConverges
centeritsatonlyConverges