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Power Series
A power series in x (or centered at 0) is a series of the following form:
1n
nnO xaa
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
A power series in x-c (or centered at c) is a series of the following
form:
1
)(n
nnO cxaa
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
Examples I
Geometric series are power series
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
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
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
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.
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
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
Examples II
Convergence of other power series
Example (1)
1
)2(
:
n
n
n
x
seriespowertheofeconvergenctheeInvestigat
),(
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
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
Conclusion
The series converges on the
interval [- ½ , ½ )
Example (2)
1 3
)2(
:
nn
nxn
seriespowertheofeconvergenctheeInvestigat
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
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
Conclusion
The series converges on the
interval (- 1 , 5 )
Example (3)
0 )!1(
)4(
:
n
n
n
x
seriespowertheofeconvergenctheeInvestigat
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
Example (4)
n
n
xn
n
seriespowertheofeconvergenctheeInvestigat
1
!
:
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
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
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
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
Hints
)2,8()5(
],()4(
)3(
)6,4[)2(
)1(
51
51
onConverges
onConverges
eveywhereabsolutelyConverges
onConverges
centeritsatonlyConverges
