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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
p
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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p
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))
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
g
0
existnotdoes
0
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If the infinite series is convergent, thenIf the infinite series is convergent, then
p
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.
p
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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Show that the following series is divergent.Show that the following series is divergent.
p
Example 3Example 3
311
1
g
!
n
n
.existnotdoes31 1limit
gp
n
n
...3333 !
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Show that the following series is divergent.Show that the following series is divergent.
p
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.
p
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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p
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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p
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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Determine if the following series is convergent or divergent.Determine if the following series is convergent or divergent.
p
Example 8Example 8
g
!2 ln
1
n nn
divergentdivergent
xxxf
ln
1)( !
g!g
dxxx2 ln
1
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Determine if the following series is convergent or divergent.Determine if the following series is convergent or divergent.
p
Example 9Example 9
g
!
0
2
n
nne
convergentconvergent
2
)( xxexf !
21
0
2
!g
dxxex
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p
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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Determine if the following series is convergent or divergent.Determine if the following series is convergent or divergent.
p
Example 10Example 10
g
! 122 cosn nn
n
divergentdivergent
nn
n 12
!
nnnn 1
cos22
"
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p
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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Determine if the following series is convergent or divergent.Determine if the following series is convergent or divergent.
p
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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p
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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p
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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End of PresentationEnd of Presentation