c# pdf free : Add link to pdf Library application component asp.net html web page mvc TRENCH_REAL_ANALYSIS20-part238

192 Chapter4
InfiniteSequencesandSeries
4.1Exercises
1.
Prove:Ifs
n
0fornkandlim
n!1
s
n
Ds,thens0.
2. (a)
Showthatlim
n!1
s
n
Ds(finite)ifandonlyiflim
n!1
js
n
sjD0.
(b)
Suppose thatjs
n
sj   t
n
forlargenand lim
n!1
t
n
D 0. Showthat
lim
n!1
s
n
Ds.
3.
Findlim
n!1
s
n
.JustifyyouranswersfromDefinition4.1.1.
(a)
s
n
D2C
1
nC1
(b)
s
n
D
˛Cn
ˇCn
(c)
s
n
D
1
n
sin
n
4
4.
Findlim
n!1
s
n
.JustifyyouranswersfromDefinition4.1.1.
(a)
s
n
D
n
2nC
p
nC1
(b)
s
n
D
n
2
C2nC2
nCn
(c)
s
n
D
sinn
p
n
(d)
s
n
D
p
n2Cnn
5.
Statenecessaryandsufficientconditionsonaconvergentsequencefs
n
gsuchthat
theintegerNinDefinition4.1.1doesnotdependupon.
6.
Prove:Iflim
n!1
s
n
Dsthenlim
n!1
js
n
jDjsj.
7.
Supposethatlim
n!1
s
n
Ds(finite)and,foreach>0,js
n
t
n
j<forlargen.
Showthatlim
n!1
t
n
Ds.
8.
CompletetheproofofTheorem4.1.6.
9.
UseTheorem4.1.6toshowthatfs
n
gconverges.
(a)
s
n
D
˛Cn
ˇCn
.ˇ>0/
(b)
s
n
D
nn
(c)
s
n
D
r
n
1Crn
.r>0/
(d)
s
n
D
.2n/Š
22n.nŠ/2
10.
LetyDTan
1
xbethesolutionofx Dtanysuchthat=2<y<=2. Prove:
Ifx
0
>0andx
nC1
DTan
1
x
n
.n0/,thenfx
n
gconverges.
11.
Supposethats
0
andAarepositivenumbers.Let
s
nC1
D
1
2
s
n
C
A
s
n
; n0:
(a)
Showthats
nC1
p
Aifn0.
(b)
Showthats
nC1
s
n
ifn1.
(c)
ShowthatsDlim
n!1
s
n
exists.
(d)
Finds.
12.
Prove:Iffs
n
gisunboundedandmonotonic,theneitherlim
n!1
s
n
D1orlim
n!1
s
n
D
1.
13.
ProveTheorem4.1.7.
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Section4.1
SequencesofRealNumbers
193
14.
UseTheorem4.1.7tofindlim
n!1
s
n
.
(a)
s
n
D
˛Cn
ˇCn
.ˇ>0/
(b)
s
n
Dcos
1
n
(c)
s
n
Dnsin
1
n
(d)
s
n
Dlognn
(e)
s
n
Dlog.nC1/log.n1/
15.
Supposethatlim
n!1
s
n
Ds(finite).Showthatifcisaconstant,thenlim
n!1
.cs
n
/D
cs.
16.
Supposethatlim
n!1
s
n
DswheresD˙1.Showthatifcisanonzeroconstant,
thenlim
n!1
.cs
n
/Dcs.
17.
Prove:Iflim
n!1
s
n
Dsandlim
n!1
t
n
Dt,wheresandtarefinite,then
lim
n!1
.s
n
Ct
n
/DsCt and
lim
n!1
.s
n
t
n
/Dst:
18.
Prove:Iflim
n!1
s
n
D sandlim
n!1
t
n
D t,wheresandtareintheextended
reals,then
lim
n!1
.s
n
Ct
n
/DsCt
ifsCtisdefined.
19.
Supposethatlim
n!1
t
n
Dt,where0<jtj<1,andlet0<<1. Showthat
thereisanintegerN suchthatt
n
>tfornN ift>0,ort
n
<tfornN if
t<0.Ineithercase,jt
n
j>jtjifnN.
20.
Prove:If
lim
n!1
s
n
s
s
n
Cs
D0; then
lim
n!1
s
n
Ds:
H
INT
:Definet
n
D.s
n
s/=.s
n
Cs/andsolvefors
n
:
21.
Prove:iflim
n!1
s
n
Dsandlim
n!1
t
n
Dt,wheresandt areintheextended
reals,then
lim
n!1
s
n
t
n
Dst
providedthatstisdefinedintheextendedreals.
22.
Prove:Iflim
n!1
s
n
Dsandlim
n!1
t
n
Dt,then
lim
n!1
s
n
t
n
D
s
t
.A/
ifs=t isdefinedintheextendedrealsandt t ¤ ¤ 0. . Giveanexamplewheres=tis
definedintheextendedplane,but(A)doesnothold.
23.
ProveTheorem4.1.9
(b)
.
24.
Find
sands
.
(a)
s
n
DŒ.1/
n
C1n
2
(b)
s
n
D.1r
n
/sin
n
2
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194 Chapter4
InfiniteSequencesandSeries
(c)
s
n
D
r2n
1Crn
.r¤1/
(d)
s
n
Dn
2
n
(e)
s
n
D.1/
n
t
n
wherelim
n!1
t
n
Dt
25.
Find
sands
.
(a)
s
n
D.1/
n
(b)
s
n
D.1/
n
2C
3
n
(c)
s
n
D
nC.1/
n
.2nC1/
n
(d)
s
n
Dsin
n
3
26.
Supposethatlim
n!1
js
n
jD(finite).Showthatfs
n
gdivergesunlessD0orthe
termsinfs
n
ghavethesamesignforlargen.H
INT
:UseExercise4.1.19:
27.
Prove:Thesequencefs
n
gisboundedif,forsomepositive,thereisanintegerN
suchthatjs
n
s
m
j<whenevern,mN.
InExercises4.1.284.1.31,assumethat
s,s
.ors/,
t,andt
areintheextendedreals,and
showthatthegiveninequalitiesorequationsholdwhenevertheirrightsidesaredefined
.notindeterminate/.
28. (a)
lim
n!1
.s
n
/Ds
(b)
lim
n!1
.s
n
/D
s
29. (a)
lim
n!1
.s
n
Ct
n
/
sC
t
(b)
lim
n!1
.s
n
Ct
n
/s
Ct
30. (a)
Ifs
n
0,t
n
0,then
(i)
lim
n!1
s
n
t
n
s
tand
(ii)
lim
n!1
s
n
t
n
s
t
.
(b)
Ifs
n
0,t
n
0,then
(i)
lim
n!1
s
n
t
n
st
and
(ii)
lim
n!1
s
n
t
n
s
t.
31. (a)
If lim
n!1
s
n
Ds>0andt
n
0,then
(i)
lim
n!1
s
n
t
n
Ds
tand
(ii)
lim
n!1
s
n
t
n
Dst
.
(b)
If lim
n!1
s
n
Ds<0andt
n
0,then
(i)
lim
n!1
s
n
t
n
Dst
and
(ii)
lim
n!1
s
n
t
n
Ds
t.
32.
Supposethatfs
n
gconvergesandhasonlyfinitelymanydistinctterms.Showthats
n
isconstantforlargen.
33.
Lets
0
ands
1
bearbitrary,and
s
nC1
D
s
n
Cs
n1
2
; n1:
UseCauchy’sconvergencecriteriontoshowthatfs
n
gconverges.
34.
Lett
n
D
s
1
Cs
2
CCs
n
n
,n1.
(a)
Prove:Iflim
n!1
s
n
Dsthenlim
n!1
t
n
Ds.
(b)
Giveanexampletoshowthatft
n
gmayconvergeeventhoughfs
n
gdoesnot.
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Section4.2
EarlierTopicsRevisitedwithSequences
195
35. (a)
Showthat
lim
n!1
1
˛
1

1
˛
2

1
˛
n
D0; if ˛>0:
H
INT
:Lookatthelogarithmoftheabsolutevalueoftheproduct:
(b)
Concludefrom
(a)
that
lim
n!1
q
n
!
D0 if q>1;
where
q
n
!
isthegeneralizedbinomialcoefficientofExample2.5.3.
4.2EARLIERTOPICSREVISITEDWITHSEQUENCES
InChapter2.3weused–ı definitionsandarguments todevelopthetheoryoflimits,
continuity,anddifferentiability;forexample,f iscontinuousatx
0
ifforeach>0there
isaı >0suchthatjf.x/f.x
0
/j <whenjxx
0
j< ı. . Thesametheorycanbe
developedbymethodsbasedonsequences. Althoughwewillnotcarrythisoutindetail,
wewilldevelopitenoughtogivesomeexamples. First,weneedanotherdefinitionabout
sequences.
Definition4.2.1
Asequenceft
k
gisasubsequenceofasequencefs
n
gif
t
k
Ds
n
k
; k0;
wherefn
k
gisanincreasinginfinitesequenceofintegersinthedomainoffs
n
g.Wedenote
thesubsequenceft
k
gbyfs
n
k
g.
Notethatfs
n
gisasubsequenceofitself,ascanbeseenbytakingn
k
Dk. Allother
subsequencesoffs
n
gareobtainedbydeletingtermsfromfs
n
gandleavingthoseremaining
intheiroriginalrelativeorder.
Example4.2.1
If
fs
n
gD
1
n
D
1;
1
2
;
1
3
;:::;
1
n
;:::
;
thenlettingn
k
D2kyieldsthesubsequence
fs
2k
gD
1
2k
D
1
2
;
1
4
;:::;
1
2k
;:::
;
andlettingn
k
D2kC1yieldsthesubsequence
fs
2kC1
gD
1
2kC1
D
1;
1
3
;:::;
1
2kC1
;:::
:
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196 Chapter4
InfiniteSequencesandSeries
Sinceasubsequencefs
n
k
gisagainasequence(withrespecttok),wemayaskwhether
fs
n
k
gconverges.
Example4.2.2
Thesequencefs
n
gdefinedby
s
n
D.1/
n
1C
1
n
doesnotconverge,butfs
n
ghassubsequencesthatdo.Forexample,
fs
2k
gD
1C
1
2k
and
lim
k!1
s
2k
D1;
while
fs
2kC1
gD
1
1
2kC1
and
lim
k!1
s
2kC1
D1:
Itcanbeshown(Exercise4.2.1)thatasubsequencefs
n
k
goffs
n
gconvergesto1ifand
onlyifn
k
isevenforksufficientlylarge,orto1ifandonlyifn
k
isoddforksufficiently
large.Otherwise,fs
n
k
gdiverges.
Thesequenceinthisexamplehassubsequencesthatconvergetodifferentlimits. The
nexttheoremshowsthatifasequenceconvergestoafinitelimitordivergesto˙1,then
allitssubsequencesdoalso.
Theorem4.2.2
If
lim
n!1
s
n
Ds .1s1/;
(4.2.1)
then
lim
k!1
s
n
k
Ds
(4.2.2)
foreverysubsequencefs
n
k
goffs
n
g:
Proof
Weconsiderthecasewheresisfiniteandleavetheresttoyou(Exercise4.2.4).
If(4.2.1)holdsand>0,thereisanintegerN suchthat
js
n
sj< if nN:
Sincefn
k
gisanincreasingsequence,thereisanintegerKsuchthatn
k
N ifk K.
Therefore,
js
n
k
Lj< if kK;
whichimplies(4.2.2).
Theorem4.2.3
Iffs
n
gismonotonicandhasasubsequencefs
n
k
gsuchthat
lim
k!1
s
n
k
Ds .1s1/;
then
lim
n!1
s
n
Ds:
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Section4.2
EarlierTopicsRevisitedwithSequences
197
Proof
Weconsiderthecasewherefs
n
gisnondecreasingandleavetheresttoyou(Ex-
ercise4.2.6).Sincefs
n
k
gisalsonondecreasinginthiscase,itsufficestoshowthat
supfs
n
k
gDsupfs
n
g
(4.2.3)
andthenapplyTheorem4.1.6
(a)
.Sincethesetoftermsoffs
n
k
giscontainedinthesetof
termsoffs
n
g,
supfs
n
gsupfs
n
k
g:
(4.2.4)
Sincefs
n
gisnondecreasing,thereisforeverynanintegern
k
suchthats
n
s
n
k
. This
impliesthat
supfs
n
g supfs
n
k
g:
Thisand(4.2.4)imply(4.2.3).
LimitPointsinTermsofSequences
InSection1.3wedefinedlimitpointintermsofneighborhoods:
xisalimitpointofaset
Sifeveryneighborhoodof
xcontainspointsofSdistinctfrom
x.Thenexttheoremshows
thatanequivalentdefinitioncanbestatedintermsofsequences.
Theorem4.2.4
Apoint
xisalimitpointofasetSifandonlyifthereisasequence
fx
n
gofpointsinSsuchthatx
n
¤
xforn1;and
lim
n!1
x
n
D
x:
Proof
Forsufficiency,supposethatthestatedconditionholds. Then,foreach  >0,
thereisanintegerNsuchthat0<jx
n
xj<ifnN.Therefore,every-neighborhood
of
xcontainsinfinitelymanypointsofS.Thismeansthat
xisalimitpointofS.
Fornecessity,let
x bealimitpointofS. . Then,foreveryintegern1,theinterval
.
x1=n;
xC1=n/containsapointx
n
x/inS. Sincejx
m
xj1=nifmn,
lim
n!1
x
n
D
x.
Wewillusethenexttheoremtoshowthatcontinuitycanbedefinedintermsofse-
quences.
Theorem4.2.5
(a)
Iffx
n
gisbounded;thenfx
n
ghasaconvergentsubsequence:
(b)
Iffx
n
gisunboundedabove;thenfx
n
ghasasubsequencefx
n
k
gsuchthat
lim
k!1
x
n
k
D1:
(c)
Iffx
n
gisunboundedbelow;thenfx
n
ghasasubsequencefx
n
k
gsuchthat
lim
k!1
x
n
k
D1:
198 Chapter4
InfiniteSequencesandSeries
Proof
Weprove
(a)
andleave
(b)
and
(c)
toyou(Exercise4.2.7). LetS S bethe
setofdistinctnumbersthatoccurastermsoffx
n
g. (Forexample, , iffx
n
g D D f.1/
n
g,
S D D f1;1g;iffx
n
g D f1;
1
2
;1;
1
3
;:::;1;1=n;:::g, S S D D f1;
1
2
;:::;1=n;:::g.) IfS
containsonlyfinitelymanypoints,thensome
xinSoccursinfinitelyofteninfx
n
g;thatis,
fx
n
ghasasubsequencefx
n
k
gsuchthatx
n
k
D
xforallk.Thenlim
k!1
x
n
k
D
x,andwe
arefinishedinthiscase.
IfSisinfinite,then,sinceSisbounded(byassumption),theBolzano–Weierstrassthe-
orem(Theorem1.3.8)impliesthatS hasalimitpoint
x. FromTheorem4.2.4,thereisa
sequenceofpointsfy
j
ginS,distinctfrom
x,suchthat
lim
j!1
y
j
D
x:
(4.2.5)
Althougheachy
j
occursasatermoffx
n
g,fy
j
gisnotnecessarilyasubsequenceoffx
n
g,
becauseifwewrite
y
j
Dx
n
j
;
thereisnoreasontoexpect thatfn
j
g isanincreasingsequence asrequiredinDefini-
tion4.2.1. However, , itisalwayspossibletopickasubsequencefn
j
k
g offn
j
gthatis
increasing,andthenthesequencefy
j
k
gDfs
n
j
k
gisasubsequenceofbothfy
j
gandfx
n
g.
Becauseof(4.2.5)andTheorem4.2.2thissubsequenceconvergesto
x.
ContinuityinTermsofSequences
Wenowshowthatcontinuitycanbedefinedandstudiedintermsofsequences.
Theorem4.2.6
Letf bedefinedonaclosedintervalŒa;bcontaining
x:Thenf is
continuousat
x.fromtherightif
xDa;fromtheleftif
xDb/ifandonlyif
lim
n!1
f.x
n
/Df.
x/
(4.2.6)
wheneverfx
n
gisasequenceofpointsinŒa;bsuchthat
lim
n!1
x
n
D
x:
(4.2.7)
Proof
Assumethata<
x<b;onlyminorchangesintheproofareneededif
xDaor
xDb.First,supposethatf iscontinuousat
xandfx
n
gisasequenceofpointsinŒa;b
satisfying(4.2.7).If>0,thereisaı>0suchthat
jf.x/f.
x/j< if jx
xj<ı:
(4.2.8)
From(4.2.7),thereisanintegerN suchthatjx
n
xj< ıifn N. . Thisand(4.2.8)
implythatjf.x
n
/f.
x/j<ifnN.Thisimplies(4.2.6),whichshowsthatthestated
conditionisnecessary.
Forsufficiency,supposethatf isdiscontinuousat
x.Thenthereisan
0
>0suchthat,
foreachpositiveintegern,thereisapointx
n
thatsatisfiestheinequality
jx
n
xj<
1
n
Section4.2
EarlierTopicsRevisitedwithSequences
199
while
jf.x
n
/f.
x/j
0
:
Thesequencefx
n
gthereforesatisfies(4.2.7),butnot(4.2.6). Hence,thestatedcondition
cannotholdiff isdiscontinuousat
x.Thisprovessufficiency.
Armedwiththetheoremswehaveprovedsofarinthissection,wecoulddevelopthe
theoryofcontinuousfunctionsbymeansofdefinitionsandproofsbasedonsequencesand
subsequences. Wegiveoneexample,anewproofofTheorem2.2.8,andleaveothersfor
exercises.
Theorem4.2.7
Iff iscontinuousonaclosedintervalŒa;b;thenf isboundedon
Œa;b:
Proof
Theproofisbycontradiction. Iff isnotboundedonŒa;b,thereisforeach
positiveintegernapointx
n
inŒa;bsuchthatjf.x
n
/j>n.Thisimpliesthat
lim
n!1
jf.x
n
/jD1:
(4.2.9)
Sincefx
n
gisbounded,fx
n
ghasaconvergentsubsequencefx
n
k
g(Theorem4.2.5
(a)
).If
xD lim
k!1
x
n
k
;
then
xisalimitpointofŒa;b,so
x2Œa;b.Iff iscontinuousonŒa;b,then
lim
k!1
f.x
n
k
/Df.
x/
byTheorem4.2.6,so
lim
k!1
jf.x
n
k
/jDjf.
x/j
(Exercise4.1.6),whichcontradicts(4.2.9). Therefore,f f cannotbebothcontinuousand
unboundedonŒa;b
4.2Exercises
1.
Lets
n
D.1/
n
.1C1=n/.Showthatlim
k!1
s
n
k
D1ifandonlyifn
k
isevenfor
largek,lim
k!1
s
n
k
D1ifandonlyifn
k
isoddforlargek,andfs
n
k
gdiverges
otherwise.
2.
FindallnumbersLintheextendedrealsthatarelimitsofsomesubsequenceoffs
n
g
and,foreachsuchL,chooseasubsequencefs
n
k
gsuchthatlim
k!1
s
n
k
DL.
(a)
s
n
D.1/
n
n
(b)
s
n
D
1C
1
n
cos
n
2
(c)
s
n
D
1
1
n2
sin
n
2
(d)
s
n
D
1
n
(e)
s
n
DŒ.1/
n
C1n
2
(f)
s
n
D
nC1
nC2
sin
n
4
Ccos
n
4
200 Chapter4
InfiniteSequencesandSeries
3.
Constructasequencefs
n
gwiththefollowingproperty,orshowthatnoneexists:for
eachpositiveintegerm,fs
n
ghasasubsequenceconvergingtom.
4.
CompletetheproofofTheorem4.2.2.
5.
Prove:Iflim
n!1
s
n
Dsandfs
n
ghasasubsequencefs
n
k
gsuchthat.1/
k
s
n
k
0,
thensD0.
6.
CompletetheproofofTheorem4.2.3.
7.
ProveTheorem4.2.5
(b)
and
(c)
.
8.
Supposethatfs
n
gisboundedandallconvergentsubsequencesoffs
n
gconvergeto
thesamelimit. Showthatfs
n
gisconvergent. Giveanexampleshowingthatthe
conclusionneednotholdiffs
n
gisunbounded.
9. (a)
Letf bedefinedonadeletedneighborhoodNof
x.Showthat
lim
x!
x
f.x/DL
ifandonlyiflim
n!1
f.x
n
/DLwheneverfx
n
gisasequenceofpointsinN
suchthatlim
n!1
x
n
D
x.H
INT
:SeetheproofofTheorem4.2.6:
(b)
Statearesultlike
(a)
forone-sidedlimits.
10.
GiveaproofbasedonsequencesforTheorem2.2.9. H
INT
:UseTheorems4.1.6;
4.2.2;4.2.5;and4.2.6:
11.
GiveaproofbasedonsequencesforTheorem2.2.12.
12.
Supposethatf is s definedonadeletedneighborhoodN N of
x andff.x
n
/g ap-
proachesalimitwheneverfx
n
g isasequenceofpointsinN N andlim
n!1
x
n
D
x. Showthatiffx
n
g andfy
n
g aretwosuchsequences, thenlim
n!1
f.x
n
/ D
lim
n!1
f.y
n
/.InferfromthisandExercise4.2.9thatlim
x!
x
f.x/exists.
13.
Prove:Iff isdefinedonaneighborhoodNof
x,thenf isdifferentiableat
xifand
onlyif
lim
n!1
f.x
n
/f.
x/
x
n
x
existswheneverfx
n
gisasequenceofpointsinNsuchthatx
n
¤
xandlim
n!1
x
n
D
x.H
INT
:UseExercise4.2.12:
4.3INFINITESERIESOFCONSTANTS
Thetheoryofsequencesdevelopedinthelasttwosectionscanbecombinedwiththefa-
miliarnotionofafinitesumtoproducethetheoryofinfiniteseries.Webeginthestudyof
infiniteseriesinthissection.
Definition4.3.1
Iffa
n
g
1
k
isaninfinitesequenceofrealnumbers,thesymbol
1
X
nDk
a
n
Section4.3
InfiniteSeriesofConstants
201
isaninfiniteseries,anda
n
isthenthtermoftheseries.Wesaythat
P
1
nDk
a
n
convergesto
thesumA,andwrite
X1
nDk
a
n
DA;
ifthesequencefA
n
g
1
k
definedby
A
n
Da
k
Ca
kC1
CCa
n
; nk;
convergestoA.ThefinitesumA
n
isthenthpartialsumof
P
1
nDk
a
n
.IffA
n
g
1
k
diverges,
wesaythat
P
1
nDk
a
n
diverges; inparticular,iflim
n!1
A
n
D 1or1,wesaythat
P
1
nDk
a
n
divergesto1or1,andwrite
X1
nDk
a
n
D1
or
X1
nDk
a
n
D1:
Adivergentinfiniteseriesthatdoesnotdivergeto˙1issaidtooscillate,orbeoscillatory.
Wewillusuallyrefertoinfiniteseriesmorebrieflyasseries.
Example4.3.1
Considertheseries
X1
nD0
r
n
; 1<r<1:
Herea
n
Dr.n0/and
A
n
D1CrCr
2
CCr
n
D
1r
nC1
1r
;
(4.3.1)
whichconvergesto1=.1r/asn!1(Example4.1.11);thus,wewrite
X1
nD0
r
n
D
1
1r
; 1<r<1:
Ifjrj>1,then(4.3.1)isstillvalid,but
P
1
nD0
r
n
diverges;ifr>1,then
X1
nD0
r
n
D1;
(4.3.2)
whileifr < < 1,
P
1
nD0
r
n
oscillates,sinceitspartialsumsalternateinsignandtheir
magnitudesbecomearbitrarilylargeforlargen.IfrD1,thenA
2mC1
D0andA
2m
D1
form0,whileifrD1,A
n
DnC1;inbothcasestheseriesdiverges,and(4.3.2)holds
ifrD1.
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