c# pdf free : Change link in pdf file SDK Library API .net wpf html sharepoint TRENCH_REAL_ANALYSIS24-part242

232 Chapter4
InfiniteSequencesandSeries
25.
Showthattheseriesconvergesabsolutely.
(a)
X
.1/
n
1
n.logn/2
(b)
X
sinn
2n
(c)
X
.1/
n
1
p
n
sin
n
(d)
X
cosn
p
n31
26.
Showthattheseriesconverges.
(a)
X
nsinn
n2C.1/n
.1<<1/
(b)
X
cosn
n
.¤2k;kD integer/
27.
Determinewhethertheseriesisabsolutelyconvergent,conditionallyconvergent,or
divergent.
(a)
X
b
n
p
n
.b
4m
Db
4mC1
D1;b
4mC2
Db
4mC3
D1/
(b)
X
1
n
sin
n
6
(c)
X
1
n2
cos
n
7
(d)
X
135.2nC1/
468.2nC4/
sinn
28.
Letgbearationalfunction(ratiooftwopolynomials).Showthat
P
g.n/rcon-
vergesabsolutelyifjrj < 1ordivergesifjrj > 1. Discussthepossibilitiesfor
jrjD1.
29.
Prove:If
P
a
2
n
<1and
P
b
2
n
<1,then
P
a
n
b
n
convergesabsolutely.
30. (a)
Prove:If
P
a
n
convergesand
P
a
2
n
D1,then
P
a
n
convergescondition-
ally.
(b)
Giveanexampleofaserieswiththepropertiesdescribedin
(a)
.
31.
Supposethat0a
nC1
<a
n
and
lim
n!1
b
1
Cb
2
CCb
n
w
n
>0;
wherefw
n
gisasequenceofpositivenumberssuchthat
X
w
n
.a
n
a
nC1
/D1:
Showthat
P
a
n
b
n
D1.H
INT
:Usesummationbyparts.
32. (a)
Prove:If0<2<<2,then
lim
n!1
jsinjCjsin2jCCjsinnj
n
sin
2
:
H
INT
:Showthatjsinnj > sinatleast“halfthetime”;moreprecisely,
showthatifjsinmjsinforsomeintegermthenjsin.mC1/j>sin.
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Section4.3
InfiniteSeriesofConstants
233
(b)
Showthat
X
sinn
np
convergesconditionallyif0<p1and¤k(kDinteger).H
INT
:Use
Exercise4.3.31andseeExample4.3.22.
33.
Showthat
X1
nD1
.1/
nC1
n
D
1
2
X1
nD1
1
n.2n1/
:
34.
Letb
3mC1
,b
3mC2
D2,andb
3mC3
D1form0.Showthat
X1
nD1
b
n
n
D
2
3
X1
mD0
1
.mC1/.3mC1/.3mC2/
:
35.
Let
P
b
n
beobtainedbyrearrangingfinitelymanytermsofaconvergentseries
P
a
n
.Showthatthetwoserieshavethesamesum.
36.
ProveTheorem4.3.26forthecasewhere
(a)
isfiniteandD1;
(b)
D1
andD1;
(c)
DD1.
37.
Givenecessaryandsufficientconditionsforadivergentseriestohaveaconvergent
rearrangement.
38.
Aseriesdivergesunconditionallyto1ifeveryrearrangementoftheseriesdiverges
to1.Statenecessaryandsufficientconditionsforaseriestohavethisproperty.
39.
Supposethatf andghavederivativesofallordersat0,andlethD D fg. Show
formallythat
X1
nD0
f
.n/
.0/
x
n
X1
nD0
g
.n/
.0/
x
n
!
D
X1
nD0
h
.n/
.0/
x
n
inthesenseoftheCauchyproduct.H
INT
:SeeExercise2.3.12.
40.
Prove:If
P
ja
n
j<1and
P
b
n
converges(perhapsconditionally),with
P
1
nD0
a
n
D
Aand
P
1
nD0
b
n
DB,thentheCauchyproduct
X1
nD0
c
n
D
X1
nD0
a
n
X1
nD0
b
n
!
convergestoAB.H
INT
:LetfA
n
g,fB
n
g,andfC
n
gbethepartialsumsoftheseries.
Showthat
C
n
A
n
BD
Xn
rD0
a
r
.B
nr
B/
andapplyTheorem4.3.5to
P
ja
n
j.
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234 Chapter4
InfiniteSequencesandSeries
41.
Supposethata
r
0forallr0andand
P
1
0
a
r
DA<1.Showthat
lim
n!1
1
n
n1
r;sD0
a
rCs
D0 and
lim
n!1
1
n
n1
r;sD0
a
rs
D2Aa
0
:
42.
Prove:Iflim
i!1
a
.i/
j
Da
j
(j 1)andja
.i/
j
j
j
(i;j 1),where
P
1
jD1
j
<
1,thenlim
i!1
P
1
jD1
a
.i/
j
D
P
1
jD1
a
j
.
43.
Prove:Ifa
n
>0,n1,and
P
1
nD1
a
n
D1,then
P
1
nD1
a
n
=.1Ca
n
/D1.
4.4SEQUENCESANDSERIESOFFUNCTIONS
Untilnowwehaveconsideredsequencesandseriesofconstants.Nowweturnourattention
tosequencesandseriesofreal-valuedfunctionsdefinedonsubsetsofthereals.Throughout
thissection,“subset”means“nonemptysubset.”
IfF
k
,F
kC1
,...,F
n
;::: arereal-valuedfunctionsdefinedonasubsetDofthereals,
wesaythatfF
n
gisaninfinitesequenceor(simplyasequence)offunctionsonD. Ifthe
sequenceofvaluesfF
n
.x/gconvergesforeachxinsomesubsetSofD,thenfF
n
gdefines
alimitfunctiononS.Theformaldefinitionisasfollows.
Definition4.4.1
SupposethatfF
n
gisasequenceoffunctionsonDandthesequence
ofvaluesfF
n
.x/gconvergesforeachx insomesubsetSofD. . ThenwesaythatfF
n
g
convergespointwiseonStothelimitfunctionF,definedby
F.x/D lim
n!1
F
n
.x/; x2S:
Example4.4.1
Thefunctions
F
n
.x/D
1
nx
nC1
n=2
; n1;
defineasequenceonDD.1;1,and
lim
n!1
F
n
.x/D
8
<
:
1; x<0;
1;
xD0;
0;
0<x1:
Therefore,fF
n
gconvergespointwiseonSDŒ0;1tothelimitfunctionFdefinedby
F.x/D
1; xD0;
0; 0<x1:
Example4.4.2
Considerthefunctions
F
n
.x/Dx
n
e
nx
; x0; n1;
(Figure4.4.1).
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Section4.4
SequencesandSeriesofFunctions
235
y
x
y =F
n
(x)=xne−nx
y = e−n
Figure4.4.1
Equatingthederivative
F
0
n
.x/Dnx
n1
e
nx
.1x/
tozeroshowsthatthemaximumvalueofF
n
.x/onŒ0;1/ise
n
,attainedatx D D 1.
Therefore,
jF
n
.x/je
n
; x0;
solim
n!1
F
n
.x/D0forallx0.Thelimitfunctioninthiscaseisidenticallyzeroon
Œ0;1/.
Example4.4.3
Forn1,letF
n
bedefinedon.1;1/by
F
n
.x/D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
0;
x<
2
n
;
n.2Cnx/; 
2
n
x<
1
n
;
n
2
x;
1
n
x<
1
n
;
n.2nx/;
1
n
x<
2
n
;
0;
x
2
n
(Figure4.4.2,page236),
SinceF
n
.0/D0foralln,lim
n!1
F
n
.0/D0.Ifx¤0,thenF
n
.x/D0ifn2=jxj.
Therefore,
lim
n!1
F
n
.x/D0; 1<x<1;
sothelimitfunctionisidenticallyzeroon.1;1/.
Example4.4.4
Foreachpositiveintegern,letS
n
bethesetofnumbersoftheform
xDp=q,wherepandqareintegerswithnocommonfactorsand1qn.Define
F
n
.x/D
1; x2S
n
;
0; x62S
n
:
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236 Chapter4
InfiniteSequencesandSeries
Ifxisirrational,thenx62S
n
foranyn,soF
n
.x/D0,n1.Ifxisrational,thenx2S
n
andF
n
.x/D1forallsufficientlylargen.Therefore,
lim
n!1
F
n
.x/DF.x/D
1 ifxisrational;
0 ifxisirrational:
y
x
y = −n
y = n
n
1
n
1
n
2
n
2
y =F
n
(x)
Figure4.4.2
UniformConvergence
Thepointwiselimitofasequenceoffunctionsmaydifferradicallyfromthefunctionsin
thesequence. InExample4.4.1, eachF
n
is continuouson.1;1, butF isnot. In
Example4.4.3,thegraphofeachF
n
hastwotriangularspikeswithheightsthattendto
1asn!1,whilethegraphofF (thex-axis)hasnone. InExample4.4.4,eachF
n
isintegrable,whileF isnonintegrableoneveryfiniteinterval. . (Exercise4.4.3). Thereis
nothinginDefinition4.4.1toprecludetheseapparentanomalies;althoughthedefinition
impliesthatforeachx
0
inS,F
n
.x
0
/approximatesF.x
0
/ifnissufficientlylarge, it
doesnotimplythatanyparticularF
n
approximatesFwelloverallofS. Toformulatea
definitionthatdoes,itisconvenienttointroducethenotation
kgk
S
Dsup
x2S
jg.x/j
andtostatethefollowinglemma.Weleavetheprooftoyou(Exercise4.4.4).
Lemma4.4.2
IfgandharedefinedonS;then
kgChk
S
kgk
S
Ckhk
S
and
kghk
S
kgk
S
khk
S
:
Moroever;ifeithergorhisboundedonS;then
kghk
S
jkgk
S
khk
S
kj:
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Section4.4
SequencesandSeriesofFunctions
237
Definition4.4.3
AsequencefF
n
goffunctionsdefinedonasetSconvergesuniformly
tothelimitfunctionFonSif
lim
n!1
jjF
n
Fk
S
D0:
Thus,fF
n
gconvergesuniformlytoFonSifforeach>0thereisanintegerNsuchthat
kF
n
Fk
S
< if nN:
(4.4.1)
IfSDŒa;bandFisthefunctionwithgraphshowninFigure4.4.3,then(4.4.1)implies
thatthegraphof
yDF
n
.x/; axb;
liesintheshadedband
F.x/<y<F.x/C; axb;
ifnN.
FromDefinition4.4.3,iffF
n
gconvergesuniformlyonS,thenfF
n
gconvergesuniformly
onanysubsetofS(Exercise4.4.6).
y
x
a
b
y =F(x) −
y =F(x) +
y =F(x)
Figure4.4.3
Example4.4.5
ThesequencefF
n
gdefinedby
F
n
.x/Dx
n
e
nx
; n1;
convergesuniformlytoF   0(thatis,totheidenticallyzerofunction)onS D Œ0;1/,
sincewesawinExample4.4.2that
kF
n
Fk
S
DkF
n
k
S
De
n
;
238 Chapter4
InfiniteSequencesandSeries
so
kF
n
Fk
S
<
ifn>log.Forthesevaluesofn,thegraphof
yDF
n
.x/; 0x<1;
liesinthestrip
y; x0
(Figure4.4.4).
Thenexttheoremprovidesalternativedefinitionsofpointwiseanduniformconvergence.
ItfollowsimmediatelyfromDefinitions4.4.1and4.4.3.
Theorem4.4.4
LetfF
n
gbedefinedonS:Then
(a)
fF
n
gconvergespointwisetoFonSifandonlyifthereis,foreach>0andx2S,
anintegerN.whichmaydependonxaswellas/suchthat
jF
n
.x/F.x/j< if nN:
(b)
fF
n
gconvergesuniformlytoFonSifandonlyifthereisforeach>0aninteger
N .whichdependsonlyonandnotonanyparticularxinS/suchthat
jF
n
.x/F.x/j< forallxinSifnN:
y
x
y = e−n
y = e
y = −e
y =xne−nx
Figure4.4.4
ThenexttheoremfollowsimmediatelyfromTheorem4.4.4andExample4.4.6.
Section4.4
SequencesandSeriesofFunctions
239
Theorem4.4.5
IffF
n
gconvergesuniformlytoFonS;thenfF
n
gconvergespointwise
toF onS:TheconverseisfalseIthatis;pointwiseconvergencedoesnotimplyuniform
convergence.
Example4.4.6
ThesequencefF
n
gofExample4.4.3convergespointwisetoF 0
on.1;1/,butnotuniformly,since
kF
n
Fk
.1;1/
DF
n
1
n
D
ˇ
ˇ
ˇ
ˇ
F
n
1
n
ˇ
ˇ
ˇ
ˇ
Dn;
so
lim
n!1
kF
n
Fk
.1;1/
D1:
However,theconvergenceisuniformon
S
D.1;[Œ;1/
forany>0,since
kF
n
Fk
S
D0 if n>
2
:
Example4.4.7
IfF
n
.x/Dx
n
,n1,thenfF
n
gconvergespointwiseonS DŒ0;1
to
F.x/D
1; xD1;
0; 0x<1:
TheconvergenceisnotuniformonS.Toseethis,supposethat0<<1.Then
jF
n
.x/F.x/j>1 if .1/
1=n
<x<1:
Therefore,
1kF
n
Fk
S
1
foralln1.Sincecanbearbitrarilysmall,itfollowsthat
kF
n
Fk
S
D1
foralln1.
However,theconvergenceisuniformonŒ0;if0<<1,sincethen
kF
n
Fk
Œ0;
D
n
andlim
n!1
n
D0. Anotherwaytosaythesamething:fF
n
gconvergesuniformlyon
everyclosedsubsetofŒ0;1/.
Thenexttheoremenablesustotestasequenceforuniformconvergencewithoutguessing
whatthelimitfunctionmightbe. ItisanalogoustoCauchy’sconvergencecriterionfor
sequencesofconstants(Theorem4.1.13).
240 Chapter4
InfiniteSequencesandSeries
Theorem4.4.6(Cauchy’sUniformConvergenceCriterion)
Asequence
offunctionsfF
n
gconvergesuniformlyonasetSifandonlyifforeach>0thereisan
integerN suchthat
kF
n
F
m
k
S
< if n;mN:
(4.4.2)
Proof
Fornecessity,supposethatfF
n
gconvergesuniformlytoFonS.Then,if>0,
thereisanintegerNsuchthat
kF
k
Fk
S
<
2
if kN:
Therefore,
kF
n
F
m
k
S
Dk.F
n
F/C.FF
m
/k
S
kF
n
Fk
S
CkFF
m
k
S
(Lemma4.4.2)
<
2
C
2
D if m;nN:
Forsufficiency,wefirstobservethat(4.4.2)impliesthat
jF
n
.x/F
m
.x/j< if n;mN;
foranyfixedxinS.Therefore,Cauchy’sconvergencecriterionforsequencesofconstants
(Theorem4.1.13)impliesthatfF
n
.x/gconvergesforeachxinS;thatis,fF
n
gconverges
pointwisetoalimitfunctionFonS.Toseethattheconvergenceisuniform,wewrite
jF
m
.x/F.x/jDjŒF
m
.x/F
n
.x/CŒF
n
.x/F.x/j
jF
m
.x/F
n
.x/jCjF
n
.x/F.x/j
kF
m
F
n
k
S
CjF
n
.x/F.x/j:
Thisand(4.4.2)implythat
jF
m
.x/F.x/j<CjF
n
.x/F.x/j if n;mN:
(4.4.3)
Sincelim
n!1
F
n
.x/DF.x/,
jF
n
.x/F.x/j<
forsomenN,so(4.4.3)impliesthat
jF
m
.x/F.x/j<2 if mN:
ButthisinequalityholdsforallxinS,so
kF
m
Fk
S
2 if mN:
Sinceisanarbitrarypositivenumber,thisimpliesthatfF
n
gconvergesuniformlytoF
onS.
ThenextexampleissimilartoExample4.1.14.
Section4.4
SequencesandSeriesofFunctions
241
Example4.4.8
SupposethatgisdifferentiableonSD.1;1/and
jg
0
.x/jr<1; 1<x<1:
(4.4.4)
LetF
0
beboundedonSanddefine
F
n
.x/Dg.F
n1
.x//; n1:
(4.4.5)
WewillshowthatfF
n
gconvergesuniformlyonS.Wefirstnotethatifuandvareanytwo
realnumbers,then(4.4.4)andthemeanvaluetheoremimplythat
jg.u/g.v/jrjuvj:
(4.4.6)
Recalling(4.4.5)andapplyingthisinequalitywithuDF
n1
.x/andvD0showsthat
jF
n
.x/jDjg.0/C.g.F
n1
.x//g.0//jjg.0/jCjg.F
n1
.x//g.0/j
jg.0/jCrjF
n1
.x/jI
therefore,sinceF
0
isboundedonS,itfollowsbyinductionthatF
n
isboundedonS for
n1. Moreover,ifn1,then(4.4.5)and(4.4.6)withuD F
n
.x/andv DF
n1
.x/
implythat
jF
nC1
.x/F
n
.x/jDjg.F
n
.x//g.F
n1
.x//jrjF
n
.x/F
n1
.x/j; 1<x<1;
so
kF
nC1
F
n
k
S
rkF
n
F
n1
k
S
:
Byinduction,thisimpliesthat
kF
nC1
F
n
k
S
r
n
kF
1
F
0
k
S
:
(4.4.7)
Ifn>m,then
kF
n
F
m
k
S
Dk.F
n
F
n1
/C.F
n1
F
n2
/CC.F
mC1
F
m
/k
S
kF
n
F
n1
k
S
CkF
n1
F
n2
k
S
CCkF
mC1
F
m
k
S
;
fromLemma4.4.2.Now(4.4.7)impliesthat
kF
n
F
m
k
S
kF
1
F
0
k
S
.1CrCr
2
CCr
nm1
/r
m
< kF
1
F
0
k
S
r
m
1r
:
Therefore,if
kF
1
F
0
k
S
r
N
1r
<;
thenkF
n
F
m
k
S
<  ifn, m  N. Therefore, , fF
n
g convergesuniformlyonS, by
Theorem4.4.6.
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