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182 Chapter4
InﬁniteSequencesandSeries
Example4.1.4
Ifs
n
D Œ1C.1/
n
n, thenfs
n
gisboundedbelow.s
n
 0/but
unboundedabove,andfs
n
gisboundedabove.s
n
0/butunboundedbelow.Ifs
n
D
.1/
n
,thenfs
n
gisbounded.Ifs
n
D.1/
n
n,thenfs
n
gisnotboundedaboveorbelow.
Theorem4.1.4
Aconvergentsequenceisbounded:
Proof
BytakingD1in(4.1.2),weseethatiflim
n!1
s
n
Ds,thenthereisaninteger
Nsuchthat
js
n
sj<1 if nN:
Therefore,
js
n
jDj.s
n
s/Csjjs
n
sjCjsj<1Cjsj if nN;
and
js
n
jmaxfjs
0
j;js
1
j;:::;js
N1
j;1Cjsjg
foralln,sofs
n
gisbounded.
MonotonicSequences
Deﬁnition4.1.5
Asequencefs
n
gisnondecreasingifs
n
 s
n1
foralln,ornonin-
creasingifs
n
s
n1
foralln:Amonotonicsequenceisasequencethatiseithernonin-
creasingornondecreasing.Ifs
n
>s
n1
foralln,thenfs
n
gisincreasing,whileifs
n
<s
n1
foralln,fs
n
gisdecreasing.
Theorem4.1.6
(a)
Iffs
n
gisnondecreasing;thenlim
n!1
s
n
Dsupfs
n
g:
(b)
Iffs
n
gisnonincreasing;thenlim
n!1
s
n
Dinffs
n
g:
Proof (a)
.LetˇDsupfs
n
g.Ifˇ<1,Theorem1.1.3impliesthatif>0then
ˇ<s
N
ˇ
forsomeintegerN.Sinces
N
s
n
ˇifnN,itfollowsthat
ˇ<s
n
ˇ if nN:
Thisimpliesthatjs
n
ˇj<ifnN,solim
n!1
s
n
Dˇ,byDeﬁnition4.1.1.IfˇD1
andbisanyrealnumber,thens
N
>bforsomeintegerN. Thens
n
>bfornN,so
lim
n!1
s
n
D1.
Weleavetheproofof
(b)
toyou(Exercise4.1.8)
Example4.1.5
Ifs
0
D1ands
n
D1es
n1
,then0<s
n
1foralln,byinduction.
Since
s
nC1
s
n
D.e
s
n
e
s
n1
/ if n1;
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Section4.1
SequencesofRealNumbers
183
themeanvaluetheorem(Theorem2.3.11)impliesthat
s
nC1
s
n
De
t
n
.s
n
s
n1
/ if n1;
(4.1.3)
wheret
n
isbetweens
n1
ands
n
.Sinces
1
s
0
D1=e<0,itfollowsbyinductionfrom
(4.1.3)thats
nC1
s
n
<0foralln.Hence,fs
n
gisboundedanddecreasing,andtherefore
convergent.
Sequences ofFunctionalValues
ThenexttheoremenablesustoapplythetheoryoflimitsdevelopedinSection2.1tosome
sequences.Weleavetheprooftoyou(Exercise4.1.13).
Theorem4.1.7
Letlim
x!1
f.x/DL;whereLisintheextendedreals;andsuppose
thats
n
Df.n/forlargen:Then
lim
n!1
s
n
DL:
Example4.1.6
Let
s
n
D
logn
n
and f.x/D
logx
x
:
ByL’Hospital’srule,
lim
x!1
logx
x
D lim
x!1
1=x
1
D0:
Hence,lim
n!1
logn=nD0.
Example4.1.7
Lets
n
D.1C1=n/
n
and
f.x/D
1C
1
x
x
De
xlog.1C1=x/
:
ByL’Hospital’srule,
lim
x!1
xlog
1C
1
x
D lim
x!1
log.1C1=x/
1=x
D lim
x!1
1
x2
1
1C1=x
1=x2
D1I
hence,
lim
x!1
1C
1
x
x
De
1
De and
lim
n!1
1C
1
n
n
De:
Thelastequationissometimesusedtodeﬁnee.
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184 Chapter4
InﬁniteSequencesandSeries
Example4.1.8
Supposethats
n
D
n
with>0,andletf.x/D
x
De
xlog
.Since
lim
x!1
e
xlog
D
8
ˆ
<
ˆ
:
0;
iflog<0 .0<<1/;
1;
iflogD0 .D1/;
1; iflog>0 .>1/;
itfollowsthat
lim
n!1
n
D
8
<
:
0;
0<<1;
1;
D1;
1; >1:
Therefore,
lim
n!1
r
n
D
8
<
:
0;
1<r<1;
1;
rD1;
1; r>1;
aresultthatwewilluseoften.
AUsefulLimitTheorem
Thenexttheoremenablesustoinvestigateconvergenceofsequencesbyexaminingsimpler
sequences.ItisanalogoustoTheorem2.1.4.
Theorem4.1.8
Let
lim
n!1
s
n
Ds and
lim
n!1
t
n
Dt;
(4.1.4)
wheresandtareﬁnite:Then
lim
n!1
.cs
n
/Dcs
(4.1.5)
ifcisaconstantI
lim
n!1
.s
n
Ct
n
/DsCt;
(4.1.6)
lim
n!1
.s
n
t
n
/Dst;
(4.1.7)
lim
n!1
.s
n
t
n
/Dst;
(4.1.8)
and
lim
n!1
s
n
t
n
D
s
t
(4.1.9)
ift
n
isnonzeroforallnandt¤0.
Proof
Weprove(4.1.8)and(4.1.9)andleavetheresttoyou(Exercises4.1.15and
4.1.17).For(4.1.8),wewrite
s
n
t
n
stDs
n
t
n
st
n
Cst
n
stD.s
n
s/t
n
Cs.t
n
t/I
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Section4.1
SequencesofRealNumbers
185
hence,
js
n
t
n
stjjs
n
sjjt
n
jCjsjjt
n
tj:
(4.1.10)
Sinceft
n
gconverges,itisbounded(Theorem4.1.4).Therefore,thereisanumberRsuch
thatjt
n
jRforalln,and(4.1.10)impliesthat
js
n
t
n
stjRjs
n
sjCjsjjt
n
tj:
(4.1.11)
From(4.1.4),if>0thereareintegersN
1
andN
2
suchthat
js
n
sj< if nN
1
(4.1.12)
and
jt
n
tj< if nN
2
:
(4.1.13)
IfN D D max.N
1
;N
2
/, then(4.1.12)and(4.1.13)bothholdwhenn  N,and(4.1.11)
impliesthat
js
n
t
n
stj.RCjsj/ if nN:
Thisproves(4.1.8).
Nowconsider(4.1.9)inthespecialcasewheres
n
D1forallnandt¤0;thus,wewant
toshowthat
lim
n!1
1
t
n
D
1
t
:
First,observethatsincelim
n!1
t
n
Dt¤0,thereisanintegerMsuchthatjt
n
jjtj=2
ifnM. Toseethis,weapplyDeﬁnition4.1.1withDjtj=2;thus,thereisaninteger
Msuchthatjt
n
tj<jt=2jifnM.Therefore,
jt
n
jDjtC.t
n
t/jjjtjjt
n
tjj
jtj
2
if nM:
If>0,chooseN
0
sothatjt
n
tj<ifnN
0
,andletN Dmax.N
0
;M/.Then
ˇ
ˇ
ˇ
ˇ
1
t
n
1
t
ˇ
ˇ
ˇ
ˇ
D
jtt
n
j
jt
n
jjtj
2
jtj2
if nNI
hence,lim
n!1
1=t
n
D1=t. Nowweobtain(4.1.9)inthegeneralcasefrom(4.1.8)with
ft
n
greplacedbyf1=t
n
g.
Example4.1.9
Todeterminethelimitofthesequencedeﬁnedby
s
n
D
1
n
sin
n
4
C
2.1C3=n/
1C1=n
;
weapplytheapplicablepartsofTheorem4.1.8asfollows:
lim
n!1
s
n
D lim
n!1
1
n
sin
n
4
C
2
h
lim
n!1
1C3lim
n!1
.1=n/
i
lim
n!1
1C lim
n!1
.1=n/
D0C
2.1C30/
1C0
D2:
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186 Chapter4
InﬁniteSequencesandSeries
Example4.1.10
Sometimespreliminarymanipulationsarenecessarybeforeapplying
Theorem4.1.8.Forexample,
lim
n!1
.n=2/Clogn
3nC4
p
n
D lim
n!1
1=2C.logn/=n
3C4n1=2
D
lim
n!1
1=2C lim
n!1
.logn/=n
lim
n!1
3C4lim
n!1
n
1=2
D
1=2C0
3C0
(seeExample4.1.6)
D
1
6
:
Example4.1.11
Supposethat1<r<1and
s
0
D1; s
1
D1Cr; s
2
D1CrCr
2
;:::; s
n
D1CrCCr
n
:
Since
s
n
rs
n
D.1CrCCr
n
/.rCr
2
CCr
nC1/
D1r
nC1
;
itfollowsthat
s
n
D
1rnC1
1r
:
(4.1.14)
FromExample4.1.8,lim
n!1
rnC1 D0,so(4.1.14)andTheorem4.1.8yield
lim
n!1
.1CrCCr
n
/D
1
1r
if 1<r<1:
Equations(4.1.5)–(4.1.8)arevalidevenifsandtarearbitraryextendedreals,provided
thattheirrightsidesaredeﬁnedintheextendedreals(Exercises4.1.16,4.1.18,and4.1.21);
(4.1.9)isvalidifs=tisdeﬁnedintheextendedrealsandt¤0(Exercise4.1.22).
Example4.1.12
If1<r<1,then
lim
n!1
r
n
D
lim
n!1
r
n
lim
n!1
D
0
1
D0;
from(4.1.9)andExample4.1.8.However,ifr>1,(4.1.9)andExample4.1.8yield
lim
n!1
r
n
D
lim
n!1
r
n
lim
n!1
D
1
1
;
anindeterminateform.Ifr1,thenlim
n!1
r
n
doesnotexistintheextendedreals,so
(4.1.9)isnotapplicable. Theorem4.1.7doesnothelpeither,sincethereisnoelementary
functionfsuchthatf.n/Dr
n
=nŠ.However,thefollowingargumentshowsthat
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Section4.1
SequencesofRealNumbers
187
lim
n!1
r
n
D0; 1<r<1:
(4.1.15)
ThereisanintegerMsuchthat
jrj
n
<
1
2
if nM:
LetKDrm=MŠ.Then
jrj
n
K
jrj
MC1
jrj
MC2

jrj
n
<K
1
2
nM
; n>M:
Given>0,chooseN MsothatK=2
NM
<. Thenjrj
n
=nŠ<ifnN,which
veriﬁes(4.1.15).
LimitsSuperiorandInferior
Requiringasequencetoconvergemaybeunnecessarilyrestrictiveinsomesituations.Of-
ten,usefulresultscanbeobtainedfromassumptionsonthelimitsuperiorandlimitinferior
ofasequence,whichweconsidernext.
Theorem4.1.9
(a)
Iffs
n
gisboundedaboveanddoesnotdivergeto1;thenthereisauniquereal
number
ssuchthat;if>0;
s
n
<
sC forlargen
(4.1.16)
and
s
n
>
s forinﬁnitelymanyn:
(4.1.17)
(b)
Iffs
n
g is boundedbelowanddoesnotdivergeto1;thenthere isauniquereal
numbers
suchthat;if>0;
s
n
>s
 forlargen
(4.1.18)
and
s
n
<s
C forinﬁnitelymanyn:
(4.1.19)
Proof
Wewillprove
(a)
andleavetheproofof
(b)
toyou(Exercise4.1.23). Since
fs
n
gisboundedabove,thereisanumberˇsuchthats
n
<ˇforalln.Sincefs
n
gdoesnot
divergeto1,thereisanumber˛suchthats
n
>˛forinﬁnitelymanyn.Ifwedeﬁne
M
k
Dsupfs
k
;s
kC1
;:::;s
kCr
;:::g;
188 Chapter4
InﬁniteSequencesandSeries
then˛M
k
ˇ,sofM
k
gisbounded.SincefM
k
gisnonincreasing(why?),itconverges,
byTheorem4.1.6.Let
sD lim
k!1
M
k
:
(4.1.20)
If>0,thenM
k
<
sCforlargek,andsinces
n
M
k
fornk,
ssatisﬁes(4.1.16).
If(4.1.17)werefalseforsomepositive,therewouldbeanintegerKsuchthat
s
n
s if nK:
However,thisimpliesthat
M
k
s if kK;
shasthestatedproperties.
Nowwemustshowthat
sistheonlyrealnumberwiththestatedproperties.Ift<
s,the
inequality
s
n
<tC
st
2
D
s
st
2
st/=2. If
s<t,theinequality
s
n
>t
t
s
2
D
sC
t
s
2
s/=2.
Therefore,
sistheonlyrealnumberwiththestatedproperties.
Deﬁnition4.1.10
Thenumbers
sands
deﬁnedinTheorem4.1.9arecalledthelimit
superiorandlimitinferior,respectively,offs
n
g,anddenotedby
sD
lim
n!1
s
n
and s
D lim
n!1
s
n
:
Wealsodeﬁne
lim
n!1
s
n
D
1
iffs
n
gisnotboundedabove;
lim
n!1
s
n
D1
if lim
n!1
s
n
D1;
lim
n!1
s
n
D1
iffs
n
gisnotboundedbelow;
and
lim
n!1
s
n
D
1
if lim
n!1
s
n
D1:
Theorem4.1.11
Everysequencefs
n
gofrealnumbershasauniquelimitsuperior;
s;
andauniquelimitinferior;s
,intheextendedreals;and
s
s:
(4.1.21)
Section4.1
SequencesofRealNumbers
189
Proof
Theexistenceanduniquenessof
sands
followfromTheorem4.1.9andDeﬁni-
tion4.1.10.If
sands
arebothﬁnite,then(4.1.16)and(4.1.18)implythat
s
<
sC
forevery>0,whichimplies(4.1.21).Ifs
D1or
sD1,then(4.1.21)isobvious.If
s
D1or
sD1,then(4.1.21)followsimmediatelyfromDeﬁnition4.1.10.
Example4.1.13
lim
n!1
r
n
D
8
<
:
1; jrj>1;
1;
jrjD1;
0;
jrj<1I
and
lim
n!1
r
n
D
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
1; r>1;
1; rD1;
0; jrj<1;
1; rD1;
1; r<1:
Also,
lim
n!1
n
2
D lim
n!1
n
2
D1;
lim
n!1
.1/
n
1
1
n
D1;
lim
n!1
.1/
n
n
1
n
D1;
and
lim
n!1
Œ1C.1/
n
n
2
D1; lim
n!1
Œ1C.1/
n
n
2
D0:
Theorem4.1.12
Iffs
n
gisasequenceofrealnumbers,then
lim
n!1
s
n
Ds
(4.1.22)
ifandonlyif
lim
n!1
s
n
D lim
n!1
s
n
Ds:
(4.1.23)
Proof
IfsD˙1,theequivalenceof(4.1.22)and(4.1.23)followsimmediatelyfrom
theirdeﬁnitions. Iflim
n!1
s
n
D s(ﬁnite),thenDeﬁnition4.1.1impliesthat(4.1.16)–
(4.1.19)holdwith
sands
replacedbys. Hence,(4.1.23)followsfromtheuniquenessof
sands
. Fortheconverse,supposethat
sDs
andletsdenotetheircommonvalue. Then
(4.1.16)and(4.1.18)implythat
s<s
n
<sC
forlargen,and(4.1.22)followsfromDeﬁnition4.1.1andtheuniquenessoflim
n!1
s
n
(Theorem4.1.2).
190 Chapter4
InﬁniteSequencesandSeries
Cauchy’sConvergenceCriterion
TodeterminefromDeﬁnition4.1.1whetherasequencehasalimit,itisnecessarytoguess
whatthelimitis. (Thisisparticularlydifﬁcultifthesequencediverges!) ) TouseTheo-
rem4.1.12forthispurposerequiresﬁnding
sands
. Thefollowingconvergencecriterion
hasneitherofthesedefects.
Theorem4.1.13(Cauchy’sConvergenceCriterion)
Asequencefs
n
gof
realnumbersconvergesifandonlyif;forevery>0;thereisanintegerNsuchthat
js
n
s
m
j< if m;nN:
(4.1.24)
Proof
Supposethatlim
n!1
s
n
Dsand>0.ByDeﬁnition4.1.1,thereisaninteger
Nsuchthat
js
r
sj<
2
if rN:
Therefore,
js
n
s
m
jDj.s
n
s/C.ss
m
/jjs
n
sjCjss
m
j< if n;mN:
Therefore,thestatedconditionisnecessaryforconvergenceoffs
n
g.Toseethatitissufﬁ-
cient,weﬁrstobservethatitimpliesthatfs
n
gisbounded(Exercise4.1.27),so
sands
are
ﬁnite(Theorem4.1.9). Nowsupposethat >0andN N satisﬁes(4.1.24). . From(4.1.16)
and(4.1.17),
js
n
sj<;
(4.1.25)
forsomeintegern>N and,from(4.1.18)and(4.1.19),
js
m
s
j<
(4.1.26)
forsomeintegerm>N.Since
j
ss
jDj.
ss
n
/C.s
n
s
m
/C.s
m
s
/j
j
ss
n
jCjs
n
s
m
jCjs
m
s
j;
(4.1.24)–(4.1.26)implythat
j
ss
j<3:
Sinceisanarbitrarypositivenumber, thisimpliesthat
s D D s
, sofs
n
gconverges, by
Theorem4.1.12.
Example4.1.14
Supposethat
jf
0
.x/jr<1; 1<x<1:
(4.1.27)
Showthattheequation
xDf.x/
(4.1.28)
hasauniquesolution.