a. Series-R Casem

8/7/2019 a. Series-R Casem

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Infinite Series of Constant Terms

Infinite Series of Positive Terms

ADVANCED CALCULUS II

by: Remalyn Quinay-Casem


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This is an example of anThis is an example of an

infinite seriesinfinite series..

1

1

Start with a square oneStart with a square one

unit by one unit:unit by one unit:

12

1

21

4

14

1

818

1

16116

1

321

64

132

164

1 !

p

Infinite SeriesInfinite Series


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In an infinite series:In an infinite series: 1 2 31

n k

k

a a a a ag

!

!

aa11, a, a22,, areare termsterms of the series.of the series. aann is theis the nnthth termterm..

Partial sums:Partial sums: 1 1S a!

2 1 2S a a!

3 1 2 3S a a a!

1

n

n k

k

S a

!

!

nnthth partial sumpartial sum

n pgp

Infinite SeriesInfinite Series


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This is an example of anThis is an example of aninfinite seriesinfinite series..

1

1

Start with a square oneStart with a square one

unit by one unit:unit by one unit:

1

21

21

4

1

4

1

81

8

1

161

16

1

321

64

1

32

1

64 1 !

The sum of the series isThe sum of the series is 11..

Many series do not converge:Many series do not converge:1 1 1 1 1

1 2 3 4 5

! g

p

This seriesThis series convergesconverges (approaches a limiting value.)(approaches a limiting value.)


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In an infinite series:In an infinite series: 1 2 31

n k

k

a a a a ag

!

!

aa11, a, a22,, areare termsterms of the series.of the series. aann is theis the nnthth termterm..

Partial sums:Partial sums: 1 1S

a!

2 1 2S a a!

3 1 2 3S a a a!

1

n

n kk

Sa!!

nnthth partial sumpartial sum

IfIfSSnn has a limit as , then the series converges,has a limit as , then the series converges,

otherwise itotherwise it divergesdiverges..

n pg

p


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Determine if the infinite series has a sum.Determine if the infinite series has a sum.

p

Example 1Example 1

g

!

g

! !

11 1

1

nn

nnn

a ...321 ! aaa

2

1

)11(1

111 !

!! as

3

2

6

1

2

1

)12(2

1

2

1212 !!!!

ass

4

3

12

1

3

2

)13(3

1

3

2323 !!

!! ass


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By partial fraction, we getBy partial fraction, we get

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Example 1Example 1

)1(1

!kk

ak

1

11

)1(

1

!

!

kkkk

ak Therefore,Therefore,

2

111 !a

3

1

2

12 !a

4

1

3

13 !a

1

11

!

nnan

Hence,Hence,

!

1

111

1

1...

4

1

3

1

3

1

2

1

2

11

nnnnsn


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Example 1Example 1

FromFrom

!

1

111

1

1...

4

1

3

1

3

1

2

1

2

11

nnnnsn

By combining terms,By combining terms,

!

1

111

1

1...

4

1

3

1

3

1

2

1

2

11

nnnnsn

we obtainwe obtain

1

11

!

nsn

!

gpgp 1

11limitlimit

ns

nn

n1!

Therefore, the series has a sum equal toTherefore, the series has a sum equal to 11..


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LetLet denotedenote aa givengiven infiniteinfinite seriesseries forfor whichwhich tt

isis thethe sequencesequence ofof partialpartial sumssums..

IfIf thethe limitlimit ofof asas nn existsexists andand isis equalequal toto

SS,, thenthen thethe seriesseries isis convergentconvergent andand SS isis thethe

sumsum ofof thethe seriesseries..

IfIf thethe limitlimit ofof asas nn doesdoes notnot exist,exist, thenthen

thethe seriesseries isis divergentdivergent andand thethe seriesseries doesdoes notnot

havehave aa sumsum..

p

DefinitionsDefinitions

g

!1nna

ns

ns

ns


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Determine if the following series is convergent orDetermine if the following series is convergent ordivergent. If it converges determine its value.divergent. If it converges determine its value.

p

ExamplesExamples

g

!1n

n ...321 !

g

! 22 1

1

n n...

15

1

8

1

3

1!

divergesdiverges

convergesconverges4

3

n

n

g

!

0

1 ...1111 ! divergesdiverges

g

!

1

1

3

1

nn ...

27

1

9

1

3

11 ! convergesconverges

2

3


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Determine the limit asDetermine the limit as nn goes to infinity of the series termsgoes to infinity of the series terms((not the partial sumsnot the partial sums))

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ExamplesExamples

g

!1n

n ...321 !

g

! 22 1

1

n n...

15

1

8

1

3

1!

divergesdiverges

convergesconverges4

3

n

n

g

!

0

1 ...1111 ! divergesdiverges

g

!

1

1

3

1

nn ...

27

1

9

1

3

11 ! convergesconverges

2

3

g

0

existnotdoes

0


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If the infinite series is convergent, thenIf the infinite series is convergent, then

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Theorem 1Theorem 1 Zero Limit TestZero Limit Test

g

!1nna 0limit !

gpn

n

a

Facts:Facts:

The converse of the theorem is not necessarily true.The converse of the theorem is not necessarily true.

It is possible to have a divergent series for whichIt is possible to have a divergent series for which 0limit !gp

nn

a

...1

...4

1

3

1

2

11

1

1

!g

! nnnharmonic seriesharmonic series..

If (or does not exist), then diverges.If (or does not exist), then diverges.g

!1n

na0limit {gp

nn

a

oneone--wayway

divergence testdivergence test


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Show that the following series is divergent.Show that the following series is divergent.

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Example 2Example 2

g

!

1

2

21

n n

n

22

2 11

1limitlimit

nn

n

nn

!

gpgp

...

16

17

9

10

4

52 !

1!


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Example 3Example 3

311

1

g

!

n

n

.existnotdoes31 1limit

gp

n

n

...3333 !


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Example 4Example 4

g

!

1 23

12

n n

n

n

n

n

n

nn 23

12

23

12limitlimit

!

gpgp

...

16

17

9

10

4

52 !

3

2!


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Determine if the following series is convergent or divergent.Determine if the following series is convergent or divergent.

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Example 4Example 4

g

!

0

3

32

210

4

n n

nn

2

1

210

43

32

0

limit !

p n

nn

ndivergent seriesdivergent series


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In aIn a geometric seriesgeometric series, each term is found by multiplying, each term is found by multiplying

the preceding term by the same number,the preceding term by the same number, rr..

2 3 1 1

1

n n

n

a ar ar ar ar ar g

!

!

This converges to if , and diverges if .This converges to if , and diverges if .1

a

r1r 1r u

1 1r is theis the interval of convergenceinterval of convergence..

p

Geometric SeriesGeometric Series

The partial sum of a geometric series is:The partial sum of a geometric series is:

11

n

n

a rS

r

!

If thenIf then1r 1

lim

1

n

n

a r

rpg

1

a

r

!

0


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3 3 3 310 100 1000 10000

.3 .03 .003 .0003 .333...!

1

3!

3

101

110

aa

rr

3

109

10

!3

9!

1

3!

p

Example 5Example 5


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1 1 112 4 8

1

11

2

1

11

2

!

1

3

2

!2

3!

aa

rr

p

Example 6Example 6


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Theorem 2Theorem 2

If converges and its sum isIf converges and its sum is SS,, then alsothen also

converges and its sum isconverges and its sum is cScS..

g

!1n

na g

!1n

nca

If diverges, then also diverges.If diverges, then also diverges.g

!1n

na g

!1nnca

c 0c 0


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Show that each of the following series are divergent.Show that each of the following series are divergent.

p

Example 7Example 7

g

!1

5

n n...

4

5

3

5

2

55 !

g

!

!1

15

n ndivergentdivergent


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p

Theorem 3Theorem 3

If and are convergent whose sums areIf and are convergent whose sums are SS

andand TT, respectively, then, respectively, then

g

!1n

na g

!1nnb

is convergent and its sum isis convergent and its sum is SS ++ TT,, g

!

1n

nn ba

is convergent and its sum isis convergent and its sum is SS TT.. g

!

1n

nn ba


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Theorem 4Theorem 4

If is convergent and is divergent, thenIf is convergent and is divergent, then

II is divergent.is divergent.

g

!1n

na g

!1n

nb

g

!

1n

nn ba

IfIf andand areare seriesseries differingdiffering onlyonly

inin their their firstfirst mm terms,terms, thenthen eithereither bothboth

convergeconverge oror bothboth divergediverge..

g

!1nna

g

!1nnb


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DefinitionDefinition

An infinite series ofAn infinite series of positive terms is convergentpositive terms is convergent iffiff itsits

sequence of partial sums has an upper bound.sequence of partial sums has an upper bound.

The is calledThe is called positive seriespositive series if everyif every

term aterm ann is positive.is positive.

g

!1n

na

Theorem 5Theorem 5


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p

Theorem 6Theorem 6 IntegralTestIntegralTest

LetLet bebe aa functionfunction thatthat isis continuous,continuous,

decreasingdecreasing andand positivepositive valuedvalued forfor allall ..

ThenThen thethe infiniteinfinite seriesseries

isis convergentconvergent ifif thethe improperimproper integralintegral aa

exists,exists, andand itit isis divergentdivergent ifif ..

1ux

g

!

!1

...)(...)3()2()1()(n

nffffnf

f

.)(limit1

g!gpb

bdxxf

g

1)( dxxf


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Example 8Example 8

g

!2 ln

1

n nn

divergentdivergent

xxxf

ln

1)( !

g!g

dxxx2 ln

1


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Example 9Example 9

g

!

0

2

n

nne

convergentconvergent

2

)( xxexf !

21

0

2

!g

dxxex


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FACTFACT pp Series TestSeries Test

IfIf k>k>00 thenthen convergesconverges ifif p>p>11 andand divergesdiverges

ifif pp11

g

!knpn

1


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Theorem 7Theorem 7 Comparison TestComparison Test

If is a convergent positive series, andIf is a convergent positive series, and

for all positive integersfor all positive integers nn, then is convergent., then is convergent.

Let be a positive series.Let be a positive series.

g

!1n

na

g

!1n

nv nn va e

g

!1n

na

If is a divergent positive series, andIf is a divergent positive series, and

for all positive integersfor all positive integers nn, then is divergent., then is divergent.

g

!1nnw nn w

a u

g

!1nna


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Example 10Example 10

g

! 122 cosn nn

n

divergentdivergent

nn

n 12

!

nnnn 1

cos22

"


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Theorem 8Theorem 8 Limit Comparison TestLimit Comparison Test

If , then the two series either bothIf , then the two series either both

converge or both diverge.converge or both diverge.

Let and be a positive series.Let and be a positive series.

g

!1nna

0limit "!gp

cb

a

n

n

n

g

!1nnb

If and if converges, thenIf and if converges, then

converges.converges.

0limit !gp

n

n

n b

a

If and if diverges, thenIf and if diverges, then

diverges.diverges.

g!gp

n

n

n b

alimit

g

!1nnb

g

!1nna

g

!1nnb

g

!1nna

n

n

n b

a

c gp! limit nnn bac

1

limit y! gp


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Example 11Example 11

g

! 0 3

1

nn n

g

!0 3

1

nn

1

3

3

1

limitn

c

n

nn

!gp 3ln3

1

1 limit nn gp!nnn

31limit ! gp 1!


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Theorem 9Theorem 9

IfIf isis aa givengiven convergentconvergent positivepositive series,series,

itsits termsterms cancan bebe groupedgrouped inin anyany manner,manner, andand

thethe resultingresulting seriesseries alsoalso willwill bebe convergentconvergent andand

willwill havehave thethe samesame sumsum asas thethe givengiven seriesseries..

g

!1n

na


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Theorem 10Theorem 10

IfIf isis aa givengiven convergentconvergent positivepositive series,series,

thethe orderorder ofof thethe termsterms cancan bebe rearranged,rearranged, andand

thethe resultingresulting seriesseries alsoalso willwill bebe convergentconvergent andand

willwill havehave thethe samesame sumsum asas thethe givengiven seriesseries..

g

!1n

na


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