Series: Guide to Investigating Convergence
A series converges to λ if the limit of the sequence of the partial sums
of the series is equal to λ
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QuestionsCheck, whether the given series is convergent, and if convergent find
its sum
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Example (2) Telescoping Series
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Examples
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Examples of this type of telescoping seriesA Convergent Telescoping Series
6
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7
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Examples of this type of telescoping seriesA Divergent Telescoping Series
)1ln(lim
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Questions ICheck, whether the given series is
convergent, and if convergent find its sum
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Questions IIShow that the following series is a telescoping series, and then determine
whether it is convergent
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The Integral Test
divergebothorconvergeboth
dxxfegralimpropertheandsseriestheeitherThen
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QuestionsCheck, whether the given series is convergent.
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Questions
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Divergence Test
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Convergence Tests for Series of Positive Terms
1. Comparison Test
2. Limit Comparison Test
3. Ratio Test
4. Root Test
The Comparison test
divergestthendivergessIf
convergessthenconvergestIf
Then
Nntsand
termspositiveofseriesbetandsLet
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Solution
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convergesn
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testcomparisonthebysoand
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DefinitionOrder of Magnitude of a Series
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tthanmagnitudeofordergreaterahass
seriesthethatsaywethent
sIf
tthanmagnitudeoforderlesserahass
seriesthethatsaywethent
sIf
magnitudeofordersamethehavetands
seriesthethatsaywethennumberpositiveaist
sIf
Then
termspositiveofseriesbetandsLet
The Limit Comparison test
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tofdivergencethetoleadssofdivergencethe
thentthanmagnitudeofordergreaterahassIf
sofeconvergencthetoleadstofeconvergencthe
thentthanmagnitudeoforderlesserahassIf
divergebothorconvergeboththeyeither
thenmagnitudeofordersamethehavetandsIf
Then
termspositiveofseriesbetandsLet
Solution
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Therefore
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nnThus
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The Ratio test
divergessthenors
sIf
convergessthens
sIf
Then
termspositiveofseriesabesLet
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1
1
1
1
1
,1lim.2
1lim.1
:
The Root test
divergessthenorsIf
convergessthensIf
Then
termspositiveofseriesabesLet
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1
1
1
1lim.2
1lim.1
:
DefinitionAlternating Series
termspositiveofsequenceaisswhere
s
Or
s
formsfollowing
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Alternating Series Convergence Test
convergestThen
toconvergentandgdecreaissIf
termspositiveofsequenceaisswhere
storstLet
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nnn
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1
1
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111
0sin
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Example
testeconvergencseriesgalternatinthebyconvergesn
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seriestheConsider
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1)1(
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1
11
DefinitionAbsolute and Conditional Convergence
divergessbutconvergesitif
llyconditionaconvergessseriesthethatsayWe
convergessif
absolutelyconvergessseriesthethatsayWe
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1
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Example (1)
llyconditionaconvergesseriesthesoand
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convergesn
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ns
seriestheConsider
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Example (2)
absolutelyconvergesn
nseriestheThus
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nseriesthesoand
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haveWe
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ns
seriestheConsider
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31
13
33
311
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?)(2sin
)1(
12sin)1(
:
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The Ratio test for Absolute Convergence
divergessthenors
sIf
absolutelyconvergessthens
sIf
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nn
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n
1
1
1
1
,1lim.2
1lim.1
Examples:Investigate the absolute convergence of
the following series
)!1(
)!12()1(.5
67
5)1(.4
)3cos(5)1(.3
!
)!12()1(.2
!
5)1(.1
1
1
3 41
1
5 91
11
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Example (1)
convergesesseriesthe
thatShowconvergesdxedxxfIegralimproperThe
nnfsand
oncontinuousandWhypositiveWhypositiveisf
haveWe
exfLet
Solution
esseriestheofeconvergenctheeInvestigat
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11
11
11
)!()(int
&
1;)(
),1[?)(?),(
:
)(
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Example (2)
convergesn
sseriesthe
thatShowconvergesdxx
dxxfIegralimproperThe
nnfsand
oncontinuousandWhyingdecreasWhypositiveisf
haveWex
xfLet
Solution
nsseriestheofeconvergenctheeInvestigat
nnn
n
nnn
12
1
12
1
2
12
1
1
1
)!(1
1)(int
&
1;)(
),1[?)(?),(
:1
1)(
:
1
1
Example (3)
2ln
1
2ln
1
ln
1]
ln
1[
lnln
1
ln
1
:
ln
1int
ln
1
2;)(
),2[?)(?),(
:ln
1)(
:
ln
1
22222
2
3
23
23
23
23
2
23
2
limlim
limlim
tx
x
dxxdx
xxdx
xx
havewe
convergesdxxx
egralimproperthe
thatshowingbyconvergesnn
thatshowwillWe
nnfsand
oncontinuousandWhyingdecreasWhypositiveisf
haveWexx
xfLet
Solution
nnsseriestheofeconvergenctheeInvestigat
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