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242 Chapter4
InfiniteSequencesandSeries
PropertiesPreservedbyUniformConvergence
Wenowstudypropertiesofthefunctionsofauniformlyconvergent sequence thatare
inheritedbythelimitfunction.Wefirstconsidercontinuity.
Theorem4.4.7
IffF
n
gconvergesuniformlytoFonSandeachF
n
iscontinuousat
apointx
0
inS;thensoisF.Similarstatementsholdforcontinuityfromtherightandleft:
Proof
SupposethateachF
n
iscontinuousatx
0
.Ifx2Sandn1,then
jF.x/F.x
0
/jjF.x/F
n
.x/jCjF
n
.x/F
n
.x
0
/jCjF
n
.x
0
/F.x
0
/j
jF
n
.x/F
n
.x
0
/jC2kF
n
Fk
S
:
(4.4.8)
Supposethat>0. SincefF
n
gconvergesuniformlytoF onS,wecanchoosensothat
kF
n
Fk
S
<.Forthisfixedn,(4.4.8)impliesthat
jF.x/F.x
0
/j<jF
n
.x/F
n
.x
0
/jC2; x2S:
(4.4.9)
SinceF
n
iscontinuousatx
0
,thereisaı>0suchthat
jF
n
.x/F
n
.x
0
/j< if jxx
0
j<ı;
so,from(4.4.9),
jF.x/F.x
0
/j<3; if jxx
0
j<ı:
Therefore,Fiscontinuousatx
0
. Similarargumentsapplytotheassertionsoncontinuity
fromtherightandleft.
Corollary4.4.8
IffF
n
gconvergesuniformlytoFonSandeachF
n
iscontinuouson
S;thensoisFIthatis;auniformlimitofcontinuousfunctionsiscontinuous.
Nowweconsiderthequestionofintegrabilityoftheuniformlimitofintegrablefunc-
tions.
Theorem4.4.9
SupposethatfF
n
gconvergesuniformlytoF onS DŒa;b.Assume
thatFandallF
n
areintegrableonŒa;b:Then
Z
b
a
F.x/dxD lim
n!1
Z
b
a
F
n
.x/dx:
(4.4.10)
Proof
Since
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
F
n
.x/dx
Z
b
a
F.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
jF
n
.x/F.x/jdx
.ba/kF
n
Fk
S
andlim
n!1
kF
n
Fk
S
D0,theconclusionfollows.
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Section4.4
SequencesandSeriesofFunctions
243
Inparticular,thistheoremimpliesthat(4.4.10)holdsifeachF
n
iscontinuousonŒa;b,
becausethenFiscontinuous(Corollary4.4.8)andthereforeintegrableonŒa;b.
ThehypothesesofTheorem4.4.9arestrongerthannecessary.Westatethenexttheorem
sothatyouwillbebetterinformedonthissubject.Weomittheproof,whichisinaccessible
ifyouskippedSection3.5,andquiteinvolvedinanycase.
Theorem4.4.10
SupposethatfF
n
gconvergespointwisetoF andeachF
n
isinte-
grableonŒa;b:
(a)
Iftheconvergenceisuniform;thenFisintegrableonŒa;band(4.4.10)holds.
(b)
IfthesequencefkF
n
k
Œa;b
gisboundedandF isintegrableonŒa;b;then(4.4.10)
holds.
Part
(a)
ofthistheoremshowsthatitisnotnecessarytoassumeinTheorem4.4.9thatF
isintegrableonŒa;b,sincethisfollowsfromtheuniformconvergence.Part
(b)
isknown
astheboundedconvergencetheorem. Neitheroftheassumptionsof
(b)
canbeomitted.
Thus,inExample4.4.3,wherefkF
n
k
Œ0;1
gisunboundedwhileFisintegrableonŒ0;1,
Z
1
0
F
n
.x/dxD1; n1; but
Z
1
0
F.x/dxD0:
InExample4.4.4,wherekF
n
k
Œa;b
D1foreveryfiniteintervalŒa;b,F
n
isintegrablefor
alln1,andF isnonintegrableoneveryinterval(Exercise4.4.3).
AfterTheorems4.4.7and4.4.9,itmayseemreasonabletoexpectthatifasequencefF
n
g
ofdifferentiablefunctionsconvergesuniformlytoF onS,thenF
0
Dlim
n!1
F
0
n
onS.
Thenextexampleshowsthatthisisnottrueingeneral.
Example4.4.9
ThesequencefF
n
gdefinedby
F
n
.x/Dx
n
sin
1
xn1
convergesuniformlytoF 0onŒr
1
;r
2
if0<r
1
<r
2
<1(or,equivalently,onevery
compactsubsetof.0;1/).However,
F
0
n
.x/Dnx
n1
sin
1
xn1
.n1/cos
1
xn1
;
sofF
0
n
.x/gdoesnotconvergeforanyxin.0;1/.
Theorem4.4.11
SupposethatF
0
n
is continuousonŒa;bforalln n  1andfF
0
n
g
convergesuniformlyonŒa;b:SupposealsothatfF
n
.x
0
/gconvergesforsomex
0
inŒa;b:
ThenfF
n
gconvergesuniformlyonŒa;btoadifferentiablelimitfunctionF;and
F
0
.x/D lim
n!1
F
0
n
.x/; a<x<b;
(4.4.11)
while
F
0
C
.a/D lim
n!1
F
0
n
.aC/ and F
0
.b/D lim
n!1
F
0
n
.b/:
(4.4.12)
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244 Chapter4
InfiniteSequencesandSeries
Proof
SinceF
0
n
iscontinuousonŒa;b,wecanwrite
F
n
.x/DF
n
.x
0
/C
Z
x
x
0
F
0
n
.t/dt; axb
(4.4.13)
(Theorem3.3.12).Nowlet
LD lim
n!1
F
n
.x
0
/
and
G.x/D lim
n!1
F
0
n
.x/:
(4.4.14)
SinceF
0
n
iscontinuousandfF
0
n
gconvergesuniformlytoGonŒa;b,Giscontinuouson
Œa;b(Corollary4.4.8);therefore,(4.4.13)andTheorem4.4.9(withFandF
n
replacedby
GandF
0
n
)implythatfF
n
gconvergespointwiseonŒa;btothelimitfunction
F.x/DLC
Z
x
x
0
G.t/dt:
(4.4.15)
TheconvergenceisactuallyuniformonŒa;b, sincesubtracting(4.4.13)from (4.4.15)
yields
jF.x/F
n
.x/jjLF
n
.x
0
/jC
ˇ
ˇ
ˇ
ˇ
Z
x
x
0
jG.t/F
0
n
.t/jdt
ˇ
ˇ
ˇ
ˇ
jLF
n
.x
0
/jCjxx
0
jkGF
0
n
k
Œa;b
;
so
kFF
n
k
Œa;b
jLF
n
.x
0
/jC.ba/kGF
0
n
k
Œa;b
;
wheretherightsideapproacheszeroasn!1.
SinceGiscontinuousonŒa;b,(4.4.14),(4.4.15),Definition2.3.6,andTheorem3.3.11
imply(4.4.11)and(4.4.12).
Infinite Series ofFunctions
InSection4.3we definedthesum ofaninfiniteseries ofconstantsas thelimitofthe
sequenceofpartialsums. Thesamedefinitioncanbeappliedtoseriesoffunctions,as
follows.
Definition4.4.12
Ifff
j
g
1
k
isasequenceofreal-valuedfunctionsdefinedonasetD
ofreals, then
P
1
jDk
f
j
isaninfiniteseries(orsimplyaseries)offunctionsonD. The
partialsumsof,
P
1
jDk
f
j
aredefinedby
F
n
D
n
X
jDk
f
j
; nk:
IffF
n
g1
k
convergespointwisetoafunctionF onasubsetS ofD,wesaythat
P
1
jDk
f
j
convergespointwisetothesumFonS,andwrite
F D
X1
jDk
f
j
; x2S:
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Section4.4
SequencesandSeriesofFunctions
245
IffF
n
gconvergesuniformlytoF onS,wesaythat
P
1
jDk
f
j
convergesuniformlytoF
onS.
Example4.4.10
Thefunctions
f
j
.x/Dx
j
; j j 0;
definetheinfiniteseries
X1
jD0
x
j
onDD.1;1/.Thenthpartialsumoftheseriesis
F
n
.x/D1CxCx
2
CCx
n
;
or,inclosedform,
F
n
.x/D
8
<
:
1x
nC1
1x
; x¤1;
nC1;
xD1
(Example4.1.11).WehaveseenearlierthatfF
n
gconvergespointwiseto
F.x/D
1
1x
ifjxj<1anddivergesifjxj1;hence,wewrite
X1
jD0
x
j
D
1
1x
; 1<x<1:
Sincethedifference
F.x/F
n
.x/D
x
nC1
1x
canbemadearbitrarilylargebytakingxcloseto1,
kFF
n
k
.1;1/
D1;
sotheconvergenceisnotuniformon.1;1/.Neitherisituniformonanyinterval.1;r
with1<r<1,since
kFF
n
k
.1;r/
1
2
foreverynoneverysuchinterval. (Why?) ) Theseriesdoesconvergeuniformlyonany
intervalŒr;rwith0<r<1,since
kFF
n
k
Œr;r
D
r
nC1
1r
andlim
n!1
r
n
D0.Putanotherway,theseriesconvergesuniformlyonclosedsubsetsof
.1;1/.
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246 Chapter4
InfiniteSequencesandSeries
Asforseriesofconstants,theconvergence,pointwiseoruniform,ofaseriesoffunctions
isnotchangedbyalteringoromittingfinitelymanyterms. Thisjustifiesadoptingthe
conventionthatweusedforseriesofconstants:whenweareinterestedonlyinwhethera
seriesoffunctionsconverges,andnotinitssum,wewillomitthelimitsonthesummation
signandwritesimply
P
f
n
.
Tests forUniformConvergenceofSeries
Theorem4.4.6iseasilyconvertedtoatheoremonuniformconvergenceofseries,asfol-
lows.
Theorem4.4.13(Cauchy’sUniformConvergenceCriterion)
Aseries
P
f
n
convergesuniformlyonasetS ifandonlyifforeach>0thereisanintegerN
suchthat
kf
n
Cf
nC1
CCf
m
k
S
< if mnN:
(4.4.16)
Proof
ApplyTheorem4.4.6tothepartialsumsof
P
f
n
,observingthat
f
n
Cf
nC1
CCf
m
DF
m
F
n1
:
SettingmD nin(4.4.16)yieldsthefollowingnecessary,butnotsufficient,condition
foruniformconvergenceofseries.ItisanalogoustoCorollary4.3.6.
Corollary4.4.14
If
P
f
n
convergesuniformlyonS;thenlim
n!1
kf
n
k
S
D0:
Theorem4.4.13leadsimmediatelytothefollowingimportanttestforuniformconver-
genceofseries.
Theorem4.4.15(Weierstrass’sTest)
The series
P
f
n
converges uniformly
onSif
kf
n
k
S
M
n
; nk;
(4.4.17)
where
P
M
n
<1:
Proof
FromCauchy’sconvergencecriterionforseriesofconstants, thereisforeach
>0anintegerNsuchthat
M
n
CM
nC1
CCM
m
< if mnN;
which,becauseof(4.4.17),impliesthat
kf
n
k
S
Ckf
nC1
k
S
CCkf
m
k
S
< if m;nN:
Lemma4.4.2andTheorem4.4.13implythat
P
f
n
convergesuniformlyonS.
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Section4.4
SequencesandSeriesofFunctions
247
Example4.4.11
TakingM
n
D1=n
2
andrecallingthat
X
1
n2
<1;
weseethat
X
1
x2Cn2
and
X
sinnx
n2
convergeuniformlyon.1;1/.
Example4.4.12
Theseries
X
f
n
.x/D
X
x
1Cx
n
convergesuniformlyonanysetSsuchthat
ˇ
ˇ
ˇ
ˇ
x
1Cx
ˇ
ˇ
ˇ
ˇ
r<1; x2S;
(4.4.18)
becauseifSissuchaset,then
kf
n
k
S
r
n
andWeierstrass’stestapplies,with
X
M
n
D
X
r
n
<1:
Since(4.4.18)isequivalentto
r
1Cr
x
r
1r
; x2S;
this means s that t the seriesconverges uniformlyonanycompact subset of.1=2;1/.
(Why?)FromCorollary4.4.14,theseriesdoesnotconvergeuniformlyonS D.1=2;b/
withb<1oronS DŒa;1/witha>1=2,becauseinthesecaseskf
n
k
S
D1forall
n.
Weierstrass’stestisveryimportant,butapplicableonlytoseriesthatactuallyexhibita
strongerkindofconvergencethanwehaveconsideredsofar.Wesaythat
P
f
n
converges
absolutelyonS if
P
jf
n
jconverges pointwiseonS, andabsolutelyuniformlyonS if
P
jf
n
jconvergesuniformlyonS. Weleaveittoyou(Exercise4.4.21)toverifythatour
proofofWeierstrass’stestactuallyshowsthat
P
f
n
convergesabsolutelyuniformlyonS.
WealsoleaveittoyoutoshowthatifaseriesconvergesabsolutelyuniformlyonS,thenit
convergesuniformlyonS(Exercise4.4.20).
Thenexttheoremappliestoseriesthatconvergeuniformly,butperhapsnotabsolutely
uniformly,onasetS.
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248 Chapter4
InfiniteSequencesandSeries
Theorem4.4.16(Dirichlet’sTestforUniformConvergence)
These-
ries
X1
nDk
f
n
g
n
convergesuniformlyonSifff
n
gconvergesuniformlytozeroonS;
P
.f
nC1
f
n
/con-
vergesabsolutelyuniformlyonS;and
kg
k
Cg
kC1
CCg
n
k
S
M; nk;
(4.4.19)
forsomeconstantM:
Proof
TheproofissimilartotheproofofTheorem4.3.20.Let
G
n
Dg
k
Cg
kC1
CCg
n
;
andconsiderthepartialsumsof
P
1
nDk
f
n
g
n
:
H
n
Df
k
g
k
Cf
kC1
g
kC1
CCf
n
g
n
:
(4.4.20)
Bysubstituting
g
k
DG
k
and g
n
DG
n
G
n1
; nkC1;
into(4.4.20),weobtain
H
n
Df
k
G
k
Cf
kC1
.G
kC1
G
k
/CCf
n
.G
n
G
n1
/;
whichwerewriteas
H
n
D.f
k
f
kC1
/G
k
C.f
kC1
f
kC2
/G
kC1
CC.f
n1
f
n
/G
n1
Cf
n
G
n
;
or
H
n
DJ
n1
Cf
n
G
n
;
(4.4.21)
where
J
n1
D.f
k
f
kC1
/G
k
C.f
kC1
f
kC2
/G
kC1
CC.f
n1
f
n
/G
n1
: (4.4.22)
Thatis,fJ
n
gisthesequenceofpartialsumsoftheseries
X1
jDk
.f
j
f
jC1
/G
j
:
(4.4.23)
From(4.4.19)andthedefinitionofG
j
,
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
Xm
jDn
Œf
j
.x/f
jC1
.x/G
j
.x/
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
M
Xm
jDn
jf
j
.x/f
jC1
.x/j; x2S;
Section4.4
SequencesandSeriesofFunctions
249
so
m
X
jDn
.f
j
f
jC1
/G
j
S
M
m
X
jDn
jf
j
f
jC1
j
S
:
Nowsupposethat>0.Since
P
.f
j
f
jC1
/convergesabsolutelyuniformlyonS,The-
orem4.4.13impliesthatthereisanintegerN suchthattherightsideofthelastinequality
islessthanifm nN. . Thesameisthentrueoftheleftside,soTheorem4.4.13
impliesthat(4.4.23)convergesuniformlyonS.
Wehave nowshownthatfJ
n
gasdefinedin(4.4.22)converges uniformlytoalimit
functionJonS.Returningto(4.4.21),weseethat
H
n
JDJ
n1
JCf
n
G
n
:
Hence,fromLemma4.4.2and(4.4.19),
kH
n
Jk
S
kJ
n1
Jk
S
Ckf
n
k
S
kG
n
k
S
kJ
n1
Jk
S
CMkf
n
k
S
:
SincefJ
n1
Jgandff
n
gconvergeuniformlytozeroonS,itnowfollowsthatlim
n!1
kH
n
Jk
S
D0.Therefore,fH
n
gconvergesuniformlyonS.
Corollary4.4.17
Theseries
P
1
nDk
f
n
g
n
convergesuniformlyonSif
f
nC1
.x/f
n
.x/; x2S; nk;
ff
n
gconvergesuniformlytozeroonS;and
kg
k
Cg
kC1
CCg
n
k
S
M; nk;
forsomeconstantM:
TheproofissimilartothatofCorollary4.3.21.Weleaveittoyou(Exercise4.4.22).
Example4.4.13
Considertheseries
X1
nD1
sinnx
n
withf
n
D1=n(constant),g
n
.x/Dsinnx,and
G
n
.x/DsinxCsin2xCCsinnx:
WesawinExample4.3.21that
jG
n
.x/j
1
jsin.x=2/j
; n1; n¤2k
(kDinteger):
250 Chapter4
InfiniteSequencesandSeries
Therefore,fkG
n
k
S
gisbounded,andtheseriesconvergesuniformlyonanysetSonwhich
sinx=2isboundedawayfromzero.Forexample,if0<ı<,then
ˇ
ˇ
ˇsin
x
2
ˇ
ˇ
ˇsin
ı
2
ifxisatleastıawayfromanymultipleof2;hence,theseriesconvergesuniformlyon
SD
[1
kD1
Œ2kCı;2.kC1/ı:
Since
X
ˇ
ˇ
ˇ
ˇ
sinnx
n
ˇ
ˇ
ˇ
ˇ
D1; x¤k
(Exercise4.3.32
(b)
),thisresultcannotbeobtainedfromWeierstrass’stest.
Example4.4.14
Theseries
X1
nD1
.1/
n
nCx2
satisfiesthehypothesesofCorollary4.4.17on.1;1/,with
f
n
.x/D
1
nCx2
; g
n
D.1/
n
; G
2m
D0;
and
G
2mC1
D1:
Therefore,theseriesconvergesuniformlyon.1;1/. Thisresultcannotbeobtainedby
Weierstrass’stest,since
X
1
nCx2
D1
forallx.
Continuity,Differentiability,andIntegrabilityofSeries
Wecanobtainresultsonthecontinuity,differentiability,andintegrabilityofinfiniteseries
byapplyingTheorems4.4.74.4.9, and4.4.11totheirpartialsums. . We e willstatethe
theoremsandgivesomeexamples,leavingtheproofstoyou.
Theorem4.4.7impliesthefollowingtheorem(Exercise4.4.23).
Theorem4.4.18
If
P
1
nDk
f
n
convergesuniformlytoF onSandeachf
n
iscontin-
uousatapointx
0
inS;thensoisF:Similarstatementsholdforcontinuityfromtheright
andleft:
Example4.4.15
InExample4.4.12wesawthattheseries
F.x/D
1
X
nD0
x
1Cx
n
Section4.4
SequencesandSeriesofFunctions
251
convergesuniformlyoneverycompactsubsetof.1=2;1/.Sincethetermsoftheseries
arecontinuousoneverysuchsubset,Theorem4.4.4impliesthatFisalso.Infact,wecan
stateastrongerresult:Fiscontinuouson.1=2;1/,sinceeverypointin.1=2;1/lies
inacompactsubintervalof.1=2;1/.
ThesameargumentandtheresultsofExample4.4.13showthatthefunction
G.x/D
X1
nD1
sinnx
n
iscontinuousexceptperhapsatx
k
D2k(kDinteger).
FromExample4.4.14,thefunction
H.x/D
X1
nD1
.1/
n
1
nCx2
iscontinuousforallx.
Thenexttheoremgivesconditionsthatpermittheinterchangeofsummationandinte-
grationofinfiniteseries. ItfollowsfromTheorem4.4.9(Exercise4.4.25). Weleaveitto
youtoformulateananalogofTheorem4.4.10forseries(Exercise4.4.26).
Theorem4.4.19
Supposethat
P
1
nDk
f
n
convergesuniformlytoF onS S D D Œa;b:
AssumethatFandf
n
;nk;areintegrableonŒa;b:Then
Z
b
a
F.x/dxD
X1
nDk
Z
b
a
f
n
.x/dx:
Wesayinthiscasethat
P
1
nDk
f
n
canbeintegratedtermbytermoverŒa;b.
Example4.4.16
FromExample4.4.10,
1
1x
D
X1
nD0
x
n
; 1<x<1:
Theseriesconvergesuniformly,andthelimitfunctionisintegrableonanyclosedsubinter-
valŒa;bof.1;1/;hence,
Z
b
a
dx
1x
D
X1
nD0
Z
b
a
x
n
dx;
so
log.1a/log.1b/D
1
X
nD0
b
nC1
a
nC1
nC1
:
LettingaD0andbDxyields
log.1x/D
X1
nD0
x
nC1
nC1
; 1<x<1:
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