c# pdf free : Adding a link to a pdf in preview Library application class asp.net html .net ajax TRENCH_REAL_ANALYSIS26-part244

252 Chapter4
InfiniteSequencesandSeries
Thenexttheoremgivesconditionsthatpermittheinterchangeofsummationanddiffer-
entiationofinfiniteseries.ItfollowsfromTheorem4.4.11(Exercise4.4.28).
Theorem4.4.20
Supposethatf
n
iscontinuouslydifferentiableonŒa;bforeachn
k;
P
1
nDk
f
n
.x
0
/convergesforsomex
0
inŒa;b;and
P
1
nDk
f
0
n
converges uniformlyon
Œa;b:Then
P
1
nDk
f
n
convergesuniformlyonŒa;btoadifferentiablefunctionF;and
F
0
.x/D
X1
nDk
f
0
n
.x/; a<x<b;
while
F
0
.aC/D
X1
nDk
f
0
n
.aC/ and F
0
.b/D
X1
nDk
f
0
n
.b/:
Wesayinthiscasethat
P
1
nDk
f
n
canbedifferentiatedtermbytermonŒa;b.Toapply
Theorem4.4.20,wefirstverifythat
P
1
nDk
f
n
.x
0
/convergesforsomex
0
inŒa;bandthen
differentiate
P
1
nDk
f
n
termbyterm.Iftheresultingseriesconvergesuniformly,thenterm
bytermdifferentiationwaslegitimate.
Example4.4.17
Theseries
X1
nD1
.1/
n
1
n
cos
x
n
(4.4.24)
convergesatx
0
D0.Differentiatingtermbytermyieldstheseries
X1
nD1
.1/
nC1
1
n2
sin
x
n
(4.4.25)
ofcontinuousfunctions. Thisseriesconvergesuniformlyon.1;1/, , byWeierstrass’s
test.ByTheorem4.4.20,theseries(4.4.24)convergesuniformlyoneveryfiniteintervalto
thedifferentiablefunction
F.x/D
X1
nD1
.1/
n
1
n
cos
x
n
; 1<x<1;
and
F
0
.x/D
X1
nD1
.1/
nC1
1
n2
sin
x
n
; 1<x<1:
Example4.4.18
Theseries
E.x/D
X1
nD0
xn
D1CxC
x2
C
x3
C
(4.4.26)
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Section4.4
SequencesandSeriesofFunctions
253
convergesuniformlyoneveryintervalŒr;rbyWeierstrass’stest,because
jxjn
rn
; jxjr;
and
X
rn
<1
forallr,bytheratiotest.Differentiatingtherightsideof(4.4.26)termbytermyieldsthe
series
X1
nD1
xn1
.n1/Š
D
X1
nD0
xn
;
whichisthesameas(4.4.26).Therefore,thedifferentiatedseriesisalsouniformlyconver-
gentonŒr;rforeveryr,sothetermbytermdifferentiationislegitimateand
E
0
.x/DE.x/; 1<x<1:
ThisisnotsurprisingifyourecognizethatE.x/Dex.
Example4.4.19
Failuretoverifythatthegivenseriesconvergesatsomepointcan
leadtoerroneousconclusions.Forexample,differentiating
X1
nD1
cos
x
n
(4.4.27)
termbytermyields
1
X
nD1
1
n
sin
x
n
;
whichconvergesuniformlyonŒr;rforeveryr,since
ˇ
ˇ
ˇ
ˇ
1
n
sin
x
n
ˇ
ˇ
ˇ
ˇ
jxj
n2
(Exercise2.3.19)
r
n2
if jxjr;
and
P
1=n
2
< 1. Wecannotconcludefromthisthat(4.4.27)convergesuniformlyon
Œr;r.Infact,itdivergesforeveryx.(Why?)
4.4Exercises
1.
FindthesetSonwhichfF
n
gconvergespointwise,andfindthelimitfunction.
(a)
F
n
.x/Dx
n
.1x
2
/
(b)
F
n
.x/Dnx
n
.1x
2
/
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254 Chapter4
InfiniteSequencesandSeries
(c)
F
n
.x/Dxn.1xn/
(d)
F
n
.x/Dsin
1C
1
n
x
(e)
F
n
.x/D
1Cxn
1Cx2n
(f)
F
n
.x/Dnsin
x
n
(g)
F
n
.x/Dn
2
1cos
x
n
(h)
F
n
.x/Dnxe
nx
2
(i)
F
n
.x/D
.xCn/
2
x2Cn2
2.
Prove:IffF
n
gconvergestoFonŒa;bandF
n
isnondecreasingforeachn,thenF
isnondecreasing.
3.
ShowthatthefunctionsfF
n
gofExample4.4.4areintegrableandFDlim
n!1
F
n
.x/
isnonintegrableoneveryfiniteinterval.
4.
ProveLemma4.4.2.
5.
FindF.x/Dlim
n!1
F
n
.x/onS. ShowthatfF
n
gconvergesuniformlytoF on
closedsubsetsofS,butnotonS.
(a)
F
n
.x/Dx
n
sinnx, SD.1;1/
(b)
F
n
.x/D
1
1Cx2n
, SDfxjx¤˙1g
(c)
F
n
.x/D
n
2
sinx
1Cn2x
, SD.0;1/H
INT
:SeeExercise2.3.19:
6. (a)
ShowthatiffF
n
gconvergesuniformlyonS,thenfF
n
gconvergesuniformly
oneverysubsetofS.
(b)
ShowthatiffF
n
gconvergesuniformlyonS
1
,S
2
,...,S
m
,thenfF
n
gcon-
vergesuniformlyon
S
m
kD1
S
k
.
(c)
GiveanexamplewherefF
n
gconvergesuniformlyoneachofaninfinitese-
quenceofsetsS
1
,S
2
,...,butnoton
S
1
kD1
S
k
.
7.
DescribethesetsonwhichthesequencesofExercise4.4.1convergeuniformly.Re-
strictyourattentiontosetsthataretheunionoffinitelymanyintervalsandsingleton
sets.
8.
SupposethatfF
n
gconvergespointwiseonŒa;band,foreachxinŒa;b,thereis
anopenintervalI
x
containingxsuchthatfF
n
gconvergesuniformlyonI
x
\Œa;b.
ShowthatfF
n
gconvergesuniformlyonŒa;b.
9.
Prove:IffF
n
gconvergesuniformlytoFonS,thenlim
n!1
kF
n
k
S
DkFk
S
.
10.
Prove:IffF
n
gconvergesuniformlytoFonS,thenF isboundedonSifandonly
if
lim
n!1
fkF
n
k
S
g<1.
11.
Prove: IffF
n
gandfG
n
gconvergeuniformlytoF andGonS,thenfF
n
CG
n
g
convergesuniformlytoFCGonS.
12. (a)
Prove:IffF
n
gandfG
n
gconvergeuniformlytoboundedfunctionsF andG
onS,thenfF
n
G
n
gconvergesuniformlytoFGonS.
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Section4.4
SequencesandSeriesofFunctions
255
(b)
Giveanexampleshowingthattheconclusionof
(a)
mayfailtoholdifF or
GisunboundedonS.
13. (a)
SupposethatfF
n
gconvergesuniformlytoFon.a;b/.Prove:Ifx
0
<a<b
andL
n
Dlim
x!x
0
F
n
.x/exists(finite)foreveryn,thenLDlim
n!1
L
n
exists(finite)and
lim
x!x
0
F.x/DL:
(b)
Statesimilarresultsforlimitsfromtherightandleft.
14.
Findthelimits.
(a)
lim
n!1
Z
4
1
n
x
sin
x
n
dx
(b)
lim
n!1
Z
2
0
dx
1Cx2n
(c)
lim
n!1
Z
1
0
nxe
nx
2
dx
(d)
lim
n!1
Z
1
0
1C
x
n
n
dx
15.
Prove(withoutusingTheorem4.4.10):IfeachF
n
isintegrableandfF
n
gconverges
uniformlyonŒa;b,thenlim
n!1
R
b
a
F
n
.x/dxexists.
16.
Prove(withoutusingTheorem4.4.10):IfeachF
n
isnondecreasingandfF
n
gcon-
vergesuniformlytoFonŒa;b,then
lim
n!1
Z
b
a
F
n
.x/dxD
Z
b
a
F.x/dx:
17.
UseWeierstrass’stesttodeterminesetsonwhichtheseriesconvergesabsolutely
uniformly.
(a)
X
1
n1=2
x
1Cx
n
(b)
X
1
n3=2
x
1Cx
n
(c)
X
nx
n
.1x/
n
(d)
X
1
n.x2Cn/
(e)
X
1
nx
(f)
X
.1x
2
/
n
.1Cx2/n
sinnx
18.
Showthatif
P
ja
n
j < 1,then
P
a
n
cosnx and
P
a
n
sinnx definecontinuous
functionson.1;1/.
19. (a)
Giveanexampleshowingthatthefollowing“comparisontest”isinvalid:If
P
f
n
convergesuniformlyonSandkg
n
k
S
 kf
n
k
S
,then
P
g
n
converges
uniformlyonS.
(b)
This“comparisontest”canbecorrectedbyaddingonewordtoitshypothesis
andconclusion.Whatistheword?
20. (a)
Explainthedifferencebetweenthefollowingstatements:
(i)
P
f
n
converges
absolutelyanduniformlyonS;
(ii)
P
f
n
converges absolutelyuniformly
onS.
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256 Chapter4
InfiniteSequencesandSeries
(b)
Showthatif
P
f
n
convergesabsolutelyuniformlyonS,then
P
f
n
converges
uniformlyonS.
21.
ShowthatthehypothesesofWeierstrass’stestimplythat
P
f
n
convergesabsolutely
uniformlyonS.
22.
ProveCorollary4.4.17.
23.
ProveTheorem4.4.18.
24.
Supposethatfa
n
g
1
1
ismonotonicandlim
n!1
a
n
D0.Showthat
X1
nD1
a
n
sinnx and
X1
nD1
a
n
cosnx
definefunctionscontinuousforallx¤2k(kDinteger).
25.
ProveTheorem4.4.19.
26.
FormulateananalogofTheorem4.4.10forseries.
27.
InSection4.5wewillseethat
e
x
2
D
X1
nD0
.1/
n
x
2n
and sinxD
X1
nD0
.1/
n
x
2nC1
.2nC1/Š
forallx,andinbothcasestheconvergenceisuniformoneveryfiniteinterval.Find
seriesthatconvergeto
(a)
F.x/D
Z
x
0
e
t
2
dt and
(b)
G.x/D
Z
x
0
sint
t
dt
forallx.
28.
ProveTheorem4.4.20.
29.
ShowfromExample4.4.17that
P
1
nD1
.1/
n
sin.x=n/convergesuniformlyonany
finiteinterval.
30.
Prove: If0 < a
nC1
< a
n
and
P
a
k
n
< 1forsomepositiveintegerk, , then
P
.1/
n
sina
n
xconvergesuniformlyonanyfiniteinterval.
31.
Forn2,define
f
n
.x/D
8
ˆ
ˆ
<
ˆ
ˆ
:
n
4
.xnC1=n
3
/; n1=n
3
xn;
n
4
.xn1=n
3
/; nxnC1=n
3
;
0;
jxnj>1=n3;
andletF.x/D
P
1
nD2
f
n
.x/.Showthat
R
1
0
F.x/dx <1,andconcludethatab-
soluteconvergenceofanimproperintegral
R
1
0
F.x/dxdoesnotimplythatlim
n!1
F.x/D
0,evenifFiscontinuousonŒ0;1/.
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Section4.5
PowerSeries
257
4.5POWERSERIES
Wenowconsideraclassofseries sufficientlygeneraltobeinteresting,butsufficiently
specializedtobeeasilyunderstood.
Definition4.5.1
Aninfiniteseriesoftheform
X1
nD0
a
n
.xx
0
/
n
;
(4.5.1)
wherex
0
anda
0
,a
1
,...,areconstants,iscalledapowerseriesinxx
0
.
Thefollowingtheoremsummarizestheconvergencepropertiesofpowerseries.
Theorem4.5.2
Inconnectionwiththepowerseries(4.5.1);defineRintheextended
realsby
1
R
D
lim
n!1
ja
n
j
1=n
:
(4.5.2)
Inparticular;RD0if
lim
n!1
ja
n
j
1=n
D1,andRD1if
lim
n!1
ja
n
j
1=n
D0:Then
thepowerseriesconverges
(a)
onlyforxDx
0
ifRD0I
(b)
forallxifRD1;andabsolutelyuniformlyineveryboundedsetI
(c)
forxin.x
0
R;x
0
CR/if0<R<1;andabsolutelyuniformlyineveryclosed
subsetofthisinterval.
Theseriesdivergesifjxx
0
j>R:Nogeneralstatementcanbemadeconcerningconver-
genceattheendpointsxDx
0
CRandxDx
0
RWtheseriesmayconvergeabsolutely
orconditionallyatboth;convergeconditionallyatoneanddivergeattheother;ordiverge
atboth:
Proof
Inanycase,theseries(4.5.1)convergestoa
0
ifxDx
0
.If
X
ja
n
jr
n
<1
(4.5.3)
forsomer > > 0, , then
P
a
n
.xx
0
/
n
convergesabsolutelyuniformlyinŒx
0
r;x
0
C
r,byWeierstrass’stest(Theorem4.4.15)andExercise4.4.21. FromCauchy’sroottest
(Theorem4.3.17),(4.5.3)holdsif
lim
n!1
.ja
n
jr
n
/
1=n
<1;
whichisequivalentto
r
lim
n!1
ja
n
j
1=n
<1
(Exercise4.1.30
(a)
). From(4.5.2), , thiscanberewrittenas r < < R,whichproves s the
assertionsconcerningconvergencein
(b)
and
(c)
.
If0R<1andjxx
0
j>R,then
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258 Chapter4
InfiniteSequencesandSeries
1
R
>
1
jxx
0
j
;
so(4.5.2)impliesthat
ja
n
j
1=n
1
jxx
0
j
andtherefore ja
n
.xx
0
/
n
j1
forinfinitelymanyvaluesofn. Therefore,
P
a
n
.xx
0
/
n
diverges(Corollary4.3.6)if
jxx
0
j>R.Inparticular,theseriesdivergesforallx¤x
0
ifRD0.
Toprovetheassertionsconcerningthepossibilitiesatx D x
0
CRandx Dx
0
R
requiresexamples,whichfollow.(Also,seeExercise4.5.1.)
ThenumberRdefinedby(4.5.2)istheradiusofconvergenceof
P
a
n
.xx
0
/n. If
R > 0,theopeninterval.x
0
R;x
0
CR/,or.1;1/ifR D 1, istheintervalof
convergenceoftheseries. Theorem4.5.2saysthatapowerserieswithanonzeroradius
ofconvergenceconvergesabsolutelyuniformlyineverycompactsubsetofitsintervalof
convergenceanddivergesateverypointintheexteriorofthisinterval.Onthislastwecan
makeastrongerstatement:Notonlydoes
P
a
n
.xx
0
/
n
divergeifjxx
0
j>R,butthe
sequencefa
n
.xx
0
/
n
gisunboundedinthiscase(Exercise4.5.3
(b)
).
Example4.5.1
Fortheseries
X
sinn=6
2n
.x1/
n
;
wehave
lim
n!1
ja
n
j
1=n
D
lim
n!1
jsinn=6
2n
1=n
D
1
2
lim
n!1
.jsinn=6j/
1=n
(Exercise4.1.30
(a)
)
D
1
2
.1/D
1
2
:
Therefore,RD2andTheorem4.5.2impliesthattheseriesconvergesabsolutelyuniformly
inclosedsubintervalsof.1;3/anddivergesifx<1orx>3.Theorem4.5.2doesnot
telluswhathappenswhenxD1orxD3,butwecanseethattheseriesdivergesinboth
thesecasessinceitsgeneraltermdoesnotapproachzero.
Example4.5.2
Fortheseries
X
x
n
n
;
lim
n!1
ja
n
j
1=n
D
lim
n!1
1
n
1=n
D
lim
n!1
exp
1
n
log
1
n
De
0
D1:
Therefore, R D D 1andtheseriesconverges absolutelyuniformlyinclosedsubintervals
of.1;1/anddivergesifjxj >1. Forx x D D 1theseriesbecomes
P
.1/
n
=n, which
convergesconditionally,andatxD1theseriesbecomes
P
1=n,whichdiverges.
Section4.5
PowerSeries
259
ThenexttheoremprovidesanexpressionforRthat,ifapplicable,isusuallyeasiertouse
than(4.5.2).
Theorem4.5.3
Theradiusofconvergenceof
P
a
n
.xx
0
/
n
isgivenby
1
R
D lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
ifthelimitexistsintheextendedreals:
Proof
FromTheorem4.5.2,itsufficestoshowthatif
LD lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
(4.5.4)
existsintheextendedreals,then
LD
lim
n!1
ja
n
j
1=n
:
(4.5.5)
Wewillshowthatthisissoif0<L<1andleavethecaseswhereLD0orLD1to
you(Exercise4.5.7).
If(4.5.4)holdswith0<L<1and0<<L,thereisanintegerNsuchthat
L<
ˇ
ˇ
ˇ
ˇ
a
mC1
a
m
ˇ
ˇ
ˇ
ˇ
<LC if mN;
so
ja
m
j.L/<ja
mC1
j<ja
m
j.LC/ if mN:
Byinduction,
ja
N
j.L/
nN
<ja
n
j<ja
N
j.LC/
nN
if n>N:
Therefore,if
K
1
Dja
N
j.L/
N
and K
2
Dja
N
j.LC/
N
;
then
K
1=n
1
.L/<ja
n
j
1=n
<K
1=n
2
.LC/:
(4.5.6)
Sincelim
n!1
K1=n D1ifKisanypositivenumber,(4.5.6)impliesthat
L lim
n!1
ja
n
j
1=n
lim
n!1
ja
n
j
1=n
LC:
Sinceisanarbitrarypositivenumber,itfollowsthat
lim
n!1
ja
n
j
1=n
DL;
whichimplies(4.5.5).
260 Chapter4
InfiniteSequencesandSeries
Example4.5.3
Forthepowerseries
X
x
n
;
lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
.nC1/Š
D lim
n!1
1
nC1
D0:
Therefore,RD1;thatis,theseriesconvergesforallx,andabsolutelyuniformlyinevery
boundedset.
Example4.5.4
Forthepowerseries
X
nŠx
n
;
lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
.nC1/Š
D lim
n!1
.nC1/D1:
Therefore,RD0,andtheseriesconvergesonlyifxD0.
Example4.5.5
Theorem4.5.3doesnotapplydirectlyto
X
.1/
n
4nnp
x
2n
(pDconstant);
(4.5.7)
whichhasinfinitelymany zerocoefficients(ofoddpowers s ofx). However, , bysetting
yDx2,weobtaintheseries
X
.1/
n
4nnp
y
n
;
(4.5.8)
whichhasnonzerocoefficientsforwhich
lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
4nnp
4nC1.nC1/p
D
1
4
lim
n!1
1C
1
n
p
D
1
4
:
Therefore, (4.5.8)convergesifjyj < < 4anddivergesifjyj > > 4. Settingy D D x
2
, we
concludethat(4.5.7)convergesifjxj<2anddivergesifjxj>2. Atx x D ˙2,(4.5.7)
becomes
P
.1/
n
=n
p
,whichdivergesifp0,convergesconditionallyif0<p1,and
convergesabsolutelyifp>1.
PropertiesofFunctions DefinedbyPowerSeries
Wenowstudythepropertiesoffunctionsdefinedbypowerseries.Henceforth,weconsider
onlypowerserieswithnonzeroradiiofconvergence.
Theorem4.5.4
Apowerseries
f.x/D
X1
nD0
a
n
.xx
0
/
n
Section4.5
PowerSeries
261
withpositiveradiusofconvergenceRiscontinuousanddifferentiableinitsintervalof
convergence;anditsderivativecanbeobtainedbydifferentiatingtermbytermIthatis;
f
0
.x/D
X1
nD1
na
n
.xx
0
/
n1
;
(4.5.9)
whichcanalsobewrittenas
f
0
.x/D
X1
nD0
.nC1/a
nC1
.xx
0
/
n
:
(4.5.10)
ThisseriesalsohasradiusofconvergenceR:
Proof
First,theseriesin(4.5.9)and(4.5.10)arethesame,sincethelatterisobtained
byshiftingtheindexofsummationintheformer.Since
lim
n!1
..nC1/ja
n
j/
1=n
D
lim
n!1
.nC1/
1=n
ja
n
j
1=n
D
lim
n!1
.nC1/
1=n

lim
n!1
ja
n
j
1=n
(Exercise4.1.30
(a)
/
D
lim
n!1
exp
log.nC1/
n

lim
n!1
ja
n
j
1=n
D
e
0
R
D
1
R
;
theradiusofconvergenceofthepowerseriesin(4.5.10)isR(Theorem4.5.2).Therefore,
thepowerseriesin(4.5.10)convergesuniformlyineveryintervalŒx
0
r;x
0
Crsuchthat
0<r<R,andTheorem4.4.20nowimplies(4.5.10)forallxin.x
0
R;x
0
CR/.
Theorem4.5.4canbestrengthenedasfollows.
Theorem4.5.5
Apowerseries
f.x/D
X1
nD0
a
n
.xx
0
/
n
withpositiveradiusofconvergenceRhasderivativesofallordersinitsintervalofconvergence;
whichcanbeobtainedbyrepeatedtermbytermdifferentiationIthus;
f
.k/
.x/D
X1
nDk
n.n1/.nkC1/a
n
.xx
0
/
nk
:
(4.5.11)
TheradiusofconvergenceofeachoftheseseriesisR:
Proof
Theproofisbyinduction. Theassertionistruefork k D1,byTheorem4.5.4.
Supposethatitistrueforsomek1.Byshiftingtheindexofsummation,wecanrewrite
(4.5.11)as
f
.k/
.x/D
1
X
nD0
.nCk/.nCk1/.nC1/a
nCk
.xx
0
/
n
; jxx
0
j<R:
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