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222 Chapter4
InfiniteSequencesandSeries
WemustshowthatADB.Supposethat>0.FromCauchy’sconvergencecriterionfor
seriesandtheabsoluteconvergenceof
P
a
n
,thereisanintegerN suchthat
ja
NC1
jCja
NC2
jCCja
NCk
j<; k1:
ChooseN
1
sothata
1
,a
2
,..., a
N
areincludedamongb
1
, b
2
, ..., , b
N
1
. Ifn   N
1
,
thenA
n
andB
n
bothincludethetermsa
1
,a
2
,...,a
N
,whichcancelonsubtraction;thus,
jA
n
B
n
jisdominatedbythesumoftheabsolutevaluesoffinitelymanytermsfrom
P
a
n
withsubscriptsgreaterthanN.Sinceeverysuchsumislessthan,
jA
n
B
n
j< if nN
1
:
Therefore,lim
n!1
.A
n
B
n
/D0andADB.
Toinvestigatetheconsequences ofrearrangingaconditionallyconvergentseries, , we
needthenexttheorem,whichisitselfimportant.
Theorem4.3.25
IfP Dfa
n
i
g
1
1
andQDfa
m
j
g
1
1
arerespectivelythesubsequences
ofallpositiveandnegativetermsinaconditionallyconvergentseries
P
a
n
;then
X1
iD1
a
n
i
D1
and
X1
jD1
a
m
j
D1:
(4.3.24)
Proof
Ifbothseriesin(4.3.24)converge,then
P
a
n
convergesabsolutely,whileifone
converges andtheotherdiverges, then
P
a
n
divergesto1or1. Hence, , bothmust
diverge.
Thenexttheoremimpliesthataconditionallyconvergentseriescanberearrangedto
produceaseriesthatconvergestoanygivennumber,divergesto˙1,oroscillates.
Theorem4.3.26
Supposethat
P
1
nD1
a
n
isconditionallyconvergentandandare
arbitrarilygivenintheextendedreals;with :Thenthetermsof
P
1
nD1
a
n
canbe
rearrangedtoformaseries
P
1
nD1
b
n
withpartialsums
B
n
Db
1
Cb
2
CCb
n
; n1;
suchthat
lim
n!1
B
n
D and
lim
n!1
B
n
D:
(4.3.25)
Proof
Weconsiderthecasewhereandarefiniteandleavetheothercasestoyou
(Exercise4.3.36).Wemayignoreanyzerotermsthatoccurin
P
1
nD1
a
n
.Forconvenience,
wedenotethepositivetermsbyPDf˛
i
g
1
1
andandthenegativetermsbyQDfˇ
j
g
1
1
.
Weconstructthesequence
fb
n
g
1
1
Df˛
1
;:::;˛
m
1
;ˇ
1
;:::;ˇ
n
1
m
1
C1
;:::;˛
m
2
;ˇ
n
1
C1
;:::;ˇ
n
2
;:::g;
(4.3.26)
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Section4.3
InfiniteSeriesofConstants
223
withsegmentschosenalternatelyfromP andQ.Letm
0
Dn
0
D0.Ifk1,letm
k
and
n
k
bethesmallestintegerssuchthatm
k
>m
k1
,n
k
>n
k1
,
m
k
X
iD1
˛
i
n
k1
X
jD1
ˇ
j
; and
m
k
X
iD1
˛
i
n
k
X
jD1
ˇ
j
:
Theorem4.3.25impliesthatthisconstructionispossible:since
P
˛
i
D
P
ˇ
j
D1,we
canchoosem
k
andn
k
sothat
m
k
X
iDm
k1
˛
i
and
n
k
X
jDn
k1
ˇ
j
areaslargeasweplease,nomatterhowlargem
k1
andn
k1
are(Exercise4.3.23).Since
m
k
andn
k
arethesmallestintegerswiththespecifiedproperties,
B
m
k
Cn
k1
<C˛
m
k
; k2;
(4.3.27)
and
ˇ
n
k
<B
m
k
Cn
k
; k2:
(4.3.28)
From(4.3.26),b
n
<0ifm
k
Cn
k1
<nm
k
Cn
k
,so
B
m
k
Cn
k
B
n
B
m
k
Cn
k1
; m
k
Cn
k1
nm
k
Cn
k
;
(4.3.29)
whileb
n
>0ifm
k
Cn
k
<nm
kC1
Cn
k
,so
B
m
k
Cn
k
B
n
B
m
kC1
Cn
k
; m
k
Cn
k
nm
kC1
Cn
k
:
(4.3.30)
Becauseof(4.3.27)and(4.3.28),(4.3.29)and(4.3.30)implythat
ˇ
n
k
<B
n
<C˛
m
k
; m
k
Cn
k1
nm
k
Cn
k
;
(4.3.31)
and
ˇ
n
k
<B
n
<C˛
m
kC1
; m
k
Cn
k
nm
kC1
Cn
k
:
(4.3.32)
Fromthefirstinequalityof(4.3.27),B
n
forinfinitelymanyvaluesofn. However,
sincelim
i!1
˛
i
D0,thesecondinequalitiesin(4.3.31)and(4.3.32)implythatif>0
thenB
n
>Cforonlyfinitelymanyvaluesofn. Therefore,
lim
n!1
B
n
D. From
thesecondinequalityin(4.3.28),B
n
forinfinitelymanyvaluesofn.However,since
lim
j!1
ˇ
j
D 0,thefirstinequalitiesin(4.3.31)and(4.3.32)implythatif > > 0then
B
n
<foronlyfinitelymanyvaluesofn.Therefore,lim
n!1
B
n
D.
MultiplicationofSeries
Theproductoftwofinitesumscanbewrittenasanotherfinitesum:forexample,
.a
0
Ca
1
Ca
2
/.b
0
Cb
1
Cb
2
/Da
0
b
0
Ca
0
b
1
Ca
0
b
2
Ca
1
b
0
Ca
1
b
1
Ca
1
b
2
Ca
2
b
0
Ca
2
b
1
Ca
2
b
2
;
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224 Chapter4
InfiniteSequencesandSeries
wherethesumontherightcontainseachproducta
i
b
j
.i;j D0;1;2/exactlyonce.These
productscan berearranged arbitrarilywithoutchangingtheirsum. Thecorresponding
situationforseriesismorecomplicated.
Giventwoseries
X1
nD0
a
n
and
X1
nD0
b
n
(becauseofapplicationsinSection4.5,itisconvenientheretostartthesummationindex
atzero),wecanarrangeallpossibleproductsa
i
b
j
.i;j 0/inatwo-dimensionalarray:
a
0
b
0
a
0
b
1
a
0
b
2
a
0
b
3

a
1
b
0
a
1
b
1
a
1
b
2
a
1
b
3

a
2
b
0
a
2
b
1
a
2
b
2
a
2
b
3

a
3
b
0
a
3
b
1
a
3
b
2
a
3
b
3

:
:
:
:
:
:
:
:
:
:
:
:
(4.3.33)
wherethesubscriptonaisconstantineachrowandthesubscriptonbisconstantineach
column.Anysensibledefinitionoftheproduct
X1
nD0
a
n
X1
nD0
b
n
!
clearlymustinvolveeveryproductinthisarrayexactlyonce;thus,wemightdefinethe
productofthetwoseriestobetheseries
P
1
nD0
p
n
,wherefp
n
gisasequenceobtained
byorderingtheproductsin(4.3.33)accordingtosomemethodthatchooseseveryproduct
exactlyonce.Onewaytodothisisindicatedby
a
0
b
0
!
a
0
b
1
a
0
b
2
!
a
0
b
3

#
"
#
a
1
b
0
a
1
b
1
a
1
b
2
a
1
b
3

#
"
#
a
2
b
0
!
a
2
b
1
!
a
2
b
2
a
2
b
3

#
a
3
b
0
a
3
b
1
a
3
b
2
a
3
b
3

#
:
:
:
:
:
:
:
:
:
:
:
:
(4.3.34)
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Section4.3
InfiniteSeriesofConstants
225
andanotherby
a
0
b
0
!
a
0
b
1
a
0
b
2
!
a
0
b
3
a
0
b
4

.
%
.
%
a
1
b
0
a
1
b
1
a
1
b
2
a
1
b
3

#
%
.
%
a
2
b
0
a
2
b
1
a
2
b
2
a
2
b
3

.
%
a
3
b
0
a
3
b
1
a
3
b
2
a
3
b
3

#
%
a
4
b
0
:
:
:
:
:
:
:
:
:
(4.3.35)
Thereareinfinitelymanyothers,andtoeachcorrespondsaseriesthatwemightconsider
tobetheproductofthegivenseries.Thisraisesaquestion:If
X1
nD0
a
n
DA and
X1
nD0
b
n
DB
whereAandBarefinite,doeseveryproductseries
P
1
nD0
p
n
constructedbyorderingthe
productsin(4.3.33)convergetoAB?
Thenexttheoremtellsuswhentheanswerisyes.
Theorem4.3.27
Let
X1
nD0
a
n
DA and
X1
nD0
b
n
DB;
whereAandBarefinite,andatleastonetermofeachseriesisnonzero.Then
P
1
nD0
p
n
D
ABforeverysequencefp
n
gobtainedbyorderingtheproductsin(4.3.33)ifandonlyif
P
a
n
and
P
b
n
convergeabsolutely:Moreover;inthiscase,
P
p
n
convergesabsolutely:
Proof
First,letfp
n
gbethesequenceobtainedbyarrangingtheproductsfa
i
b
j
gaccord-
ingtotheschemeindicatedin(4.3.34),anddefine
A
n
Da
0
Ca
1
CCa
n
;
A
n
Dja
0
jCja
1
jCCja
n
j;
B
n
Db
0
Cb
1
CCb
n
;
B
n
Djb
0
jCjb
1
jCCjb
n
j;
P
n
Dp
0
Cp
1
CCp
n
;
P
n
Djp
0
jCjp
1
jCCjp
n
j:
From(4.3.34),weseethat
P
0
DA
0
B
0
; P
3
DA
1
B
1
; P
8
DA
2
B
2
;
and,ingeneral,
P
.mC1/21
DA
m
B
m
:
(4.3.36)
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226 Chapter4
InfiniteSequencesandSeries
Similarly,
P
.mC1/21
D
A
m
B
m
:
(4.3.37)
If
P
ja
n
j<1and
P
jb
n
j<1,thenf
A
m
B
m
gisboundedand,since
P
m
P
.mC1/21
,
(4.3.37)impliesthatf
P
m
gisbounded.Therefore,
P
jp
n
j<1,so
P
p
n
converges.Now
X1
nD0
p
n
D lim
n!1
P
n
(bydefinition)
D lim
m!1
P
.mC1/
2
1
(byTheorem4.2.2)
D lim
m!1
A
m
B
m
(from(4.3.36))
D
lim
m!1
A
m

lim
m!1
B
m
(byTheorem4.1.8)
DAB:
Sinceanyotherorderingoftheproductsin(4.3.33)producesaarearrangementofthe
absolutelyconvergentseries
P
1
nD0
p
n
,Theorem4.3.24impliesthat
P
jq
n
j<1forevery
suchorderingandthat
P
1
nD0
q
n
DAB.Thisshowsthatthestatedconditionissufficient.
Fornecessity,againlet
P
1
nD0
p
n
beobtainedfromtheorderingindicatedin(4.3.34),
andsupposethat
P
1
nD0
p
n
andallitsrearrangementsconvergetoAB. Then
P
p
n
must
convergeabsolutely,byTheorem4.3.26. Therefore, , f
P
m
2
1
gisbounded,and(4.3.37)
impliesthatf
A
m
gandf
B
m
garebounded.(Hereweneedtheassumptionthatneither
P
a
n
nor
P
b
n
consistsentirelyofzeros.Why?)Therefore,
P
ja
n
j<1and
P
jb
n
j<1.
ThefollowingdefinitionoftheproductoftwoseriesisduetoCauchy. Wewillseethe
importanceofthisdefinitioninSection4.5.
Definition4.3.28
TheCauchyproductof
P
1
nD0
a
n
and
P
1
nD0
b
n
is
P
1
nD0
c
n
,where
c
n
Da
0
b
n
Ca
1
b
n1
CCa
n1
b
1
Ca
n
b
0
:
(4.3.38)
Thus,c
n
isthesumofallproductsa
i
b
j
,wherei0,j 0,andiCj Dn;thus,
c
n
D
n
X
rD0
a
r
b
nr
D
n
X
rD0
b
r
a
nr
:
(4.3.39)
Henceforth,
P
1
nD0
a
n
P
1
nD0
b
n
shouldbeinterpretedastheCauchyproduct.Notice
that
X1
nD0
a
n
X1
nD0
b
n
!
D
X1
nD0
b
n
X1
nD0
a
n
!
;
andthattheCauchyproductoftwoseriesisdefinedevenifoneorbothdiverge.Inthecase
wherebothconverge,itisnaturaltoinquireabouttherelationshipbetweentheproductof
theirsumsandthesumoftheCauchyproduct. Theorem4.3.27yieldsapartialanswerto
thisquestion,asfollows.
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Section4.3
InfiniteSeriesofConstants
227
Theorem4.3.29
If
P
1
nD0
a
n
and
P
1
nD0
b
n
convergeabsolutelytosumsAandB;
thentheCauchyproductof
P
1
nD0
a
n
and
P
1
nD0
b
n
convergesabsolutelytoAB:
Proof
LetC
n
bethenthpartialsumoftheCauchyproduct;thatis,
C
n
Dc
0
Cc
1
CCc
n
(see(4.3.38)). Let
P
1
nD0
p
n
betheseriesobtainedbyorderingtheproductsfa
i
;b
j
gac-
cordingtotheschemeindicatedin(4.3.35),anddefineP
n
tobeitsnthpartialsum;thus,
P
n
Dp
0
Cp
1
CCp
n
:
Inspectionof(4.3.35)showsthatc
n
isthesumofthenC1termsconnectedbythediagonal
arrows.Therefore,C
n
DP
m
n
,where
m
n
D1C2CC.nC1/1D
n.nC3/
2
:
FromTheorem4.3.27,lim
n!1
P
m
n
DAB,solim
n!1
C
n
DAB.Toseethat
P
jc
n
j<
1,weobservethat
Xn
rD0
jc
r
j
m
n
X
sD0
jp
s
j
andrecallthat
P
jp
s
j<1,fromTheorem4.3.27.
Example4.3.25
ConsidertheCauchyproductof
P
1
nD0
rwithitself. Herea
n
D
b
n
Dr
n
and(4.3.39)yields
c
n
Dr
0
r
n
Cr
1
r
n1
CCr
n1
r
1
Cr
n
r
0
D.nC1/r
n
;
so
1
X
nD0
r
n
!
2
D
1
X
nD0
.nC1/r
n
:
Since
X1
nD0
r
n
D
1
1r
; jrj<1;
andtheconvergenceisabsolute,Theorem4.3.29impliesthat
X1
nD0
.nC1/r
n
D
1
.1r/2
; jrj<1:
Example4.3.26
If
X1
nD0
a
n
D
X1
nD0
˛n
and
X1
nD0
b
n
D
X1
nD0
ˇn
;
228 Chapter4
InfiniteSequencesandSeries
then(4.3.39)yields
c
n
D
Xn
mD0
˛
nm
ˇ
m
.nm/ŠmŠ
D
1
Xn
mD0
n
m
!
˛
nm
ˇ
m
D
.˛Cˇ/
n
I
thus,
X1
nD0
˛
n
X1
nD0
ˇ
n
!
D
X1
nD0
.˛Cˇ/
n
:
(4.3.40)
Youprobablyknowfromcalculusthat
P
1
nD0
x
n
=nŠconvergesabsolutelyforallxtoe
x
.
Thus,(4.3.40)impliesthat
e
˛
e
ˇ
De
˛Cˇ
;
afamiliarresult.
TheCauchyproductoftwoseriesmayconvergeunderconditionsweakerthanthose
ofTheorem 4.3.29. Ifoneseries s converges absolutelyandtheotherconverges condi-
tionally,theCauchyproductofthetwoseriesconvergestotheproductofthetwosums
(Exercise4.3.40). IftwoseriesandtheirCauchyproductallconverge, , thenthesumof
theCauchyproductequalstheproductofthesumsofthetwoseries(Exercise4.5.32).
However,thenextexampleshowsthattheCauchyproductoftwoconditionallyconvergent
seriesmaydiverge.
Example4.3.27
If
a
n
Db
n
D
.1/
nC1
p
nC1
;
then
P
1
nD0
a
n
and
P
1
nD0
b
n
convergeconditionally. From(4.3.39),thegeneraltermof
theirCauchyproductis
c
n
D
Xn
rD0
.1/rC1.1/nrC1
p
rC1
p
nrC1
D.1/
n
Xn
rD0
1
p
rC1
1
p
nrC1
;
so
jc
n
j
Xn
rD0
1
p
nC1
1
p
nC1
D
nC1
nC1
D1:
Therefore,theCauchyproductdiverges,byCorollary4.3.6.
4.3Exercises
1.
ProveTheorem4.3.2.
2.
ProveTheorem4.3.3.
3. (a)
Prove:Ifa
n
Db
n
exceptforfinitelymanyvaluesofn,then
P
a
n
and
P
b
n
convergeordivergetogether.
Section4.3
InfiniteSeriesofConstants
229
(b)
Letb
n
k
Da
k
forsomeincreasingsequencefn
k
g
1
1
ofpositiveintegers,and
b
n
D0ifnisanyotherpositiveinteger.Showthat
X1
nD1
b
n
and
X1
nD1
a
n
divergeorconvergetogether,andthatinthelattercasetheyhavethesamesum.
(Thus,theconvergencepropertiesofaseriesarenotchangedbyinsertingzeros
betweenitsterms.)
4. (a)
Prove:If
P
a
n
converges,then
lim
n!1
.a
n
Ca
nC1
CCa
nCr
/D0; r0:
(b)
Does
(a)
implythat
P
a
n
converges?Giveareasonforyouranswer.
5.
ProveCorollary4.3.7.
6. (a)
VerifyCorollary4.3.7fortheconvergentseries
P
1=n
p
.p>1/. H
INT
:See
theproofofTheorem4.3.10:
(b)
VerifyCorollary4.3.7fortheconvergentseries
P
.1/
n
=n.
7.
Prove:If0b
n
a
n
b
nC1
,then
P
a
n
and
P
b
n
convergeordivergetogether.
8.
Determineconvergenceordivergence.
(a)
X
p
n1
p
n5C1
(b)
X
1
n2
1C
1
2
sin.n=4/
(c)
X
1e
n
logn
n
(d)
X
cos
n2
(e)
X
sin
n2
(f)
X
1
n
tan
n
(g)
X
1
n
cot
n
(h)
X
logn
n2
9.
Supposethatf.x/0forxk.Provethat
R
1
k
f.x/dx<1ifandonlyif
X1
nDk
Z
nC1
n
f.x/dx<1:
H
INT
:UseTheorems3.4.5and4.3.8.
10.
Usetheintegraltesttofindallvaluesofpforwhichtheseriesconverges.
(a)
X
n
.n21/p
(b)
X
n
2
.nC4/p
(c)
X
sinhn
.coshn/p
230 Chapter4
InfiniteSequencesandSeries
11.
LetL
n
bethenthiteratedlogarithm.Showthat
X
1
L
0
.n/L
1
.n/L
k
.n/ŒL
kC1
.n/
p
convergesifandonlyifp>1.H
INT
:SeeExercise3.4.10.
12.
Supposethatg,g
0
,and.g
0
/
2
gg
00
areallpositiveonŒR;1/.Showthat
X
g0.n/
g.n/
<1
ifandonlyiflim
x!1
g.x/<1.
13.
Let
S.p/D
X1
nD1
1
np
; p>1:
Showthat
1
.p1/.NC1/p1
<S.p/
N
X
nD1
1
np
<
1
.p1/Np1
:
H
INT
:SeetheproofofTheorem4.3.10.
14.
Supposethatf ispositive,decreasing,andlocallyintegrableonŒ1;1,andlet
a
n
D
Xn
kD1
f.k/
Z
n
1
f.x/dx:
(a)
Showthatfa
n
gisnonincreasingandnonnegative,and
0< lim
n!1
a
n
<f.1/:
(b)
Deducefrom
(a)
that
D lim
n!1
1C
1
2
C
1
3
CC
1
n
logn
exists,and0<<1.(isEuler’sconstant;0:577.)
15.
Determineconvergenceordivergence.
(a)
X
2Csinn
n2Csinn
(b)
X
nC1
n
r
n
.r>0/
(c)
X
e
n
coshn.>0/
(d)
X
nClogn
n2.logn/2
(e)
X
nClogn
n2logn
(f)
X
.1C1=n/
n
2n
Section4.3
InfiniteSeriesofConstants
231
16.
LetL
n
bethenthiteratedlogarithm.Provethat
X
1
ŒL
0
.n/
q
0
C1
ŒL
1
.n/
q
1
C1
ŒL
m
.n/
q
m
C1
convergesifandonlyifthereisatleastonenonzeronumberinfq
0
;q
1
;:::;q
m
gand
thefirstsuchispositive.H
INT
: SeeExercises4.3.11and2.4.42.b/:
17.
Determineconvergenceordivergence.
(a)
X
2Csin
2
.n=4/
3n
(b)
X
n.nC1/
4n
(c)
X
3sin.n=2/
n.nC1/
(d)
X
nC.1/
n
n.nC1/
18.
Determineconvergenceordivergence,withr>0.
(a)
X
rn
(b)
X
n
p
r
n
(c)
X
r
n
(d)
X
r
2nC1
.2nC1/Š
(e)
X
r
2n
.2n/Š
19.
Determineconvergenceordivergence.
(a)
X
.2n/Š
22n.nŠ/2
(b)
X
.3n/Š
33nnŠ.nC1/Š.nC3/Š
(c)
X
2
n
57.2nC3/
(d)
X
˛.˛C1/.˛Cn1/
ˇ.ˇC1/.ˇCn1/
.˛;ˇ>0/
20.
Determineconvergenceordivergence.
(a)
X
n
n
.2C.1/
n
/
2n
(b)
X
1Csin3n
3
n
(c)
X
.nC1/
1Csin.n=6/
3
n
(d)
X
1
1
n
n
2
21.
Givecounterexamplesshowingthatthefollowingstatementsarefalseunlessitis
assumedthatthetermsoftheserieshavethesamesignfornsufficientlylarge.
(a)
P
a
n
convergesifitspartialsumsarebounded.
(b)
Ifb
n
¤0fornkandlim
n!1
a
n
=b
n
DL,where0<L<1,then
P
a
n
and
P
b
n
convergeordivergetogether.
(c)
Ifa
n
¤0and
lim
n!1
a
nC1
=a
n
<1,then
P
a
n
converges.
(d)
Ifa
n
¤0and
lim
n!1
nŒ.a
nC1
=a
n
/1<1,then
P
a
n
converges.
22.
Prove:Ifthetermsofaconvergentseries
P
a
n
havethesamesignfornk,then
P
a
n
convergesabsolutely.
23.
Supposethata
n
0fornmand
P
a
n
D1.Prove:IfNisanarbitraryinteger
mandJisanarbitrarypositivenumber,then
P
NCk
nDN
a
n
>Jforsomepositive
integerk.
24.
ProveTheorem4.3.19.
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