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272 Chapter4
InfiniteSequencesandSeries
Example4.5.15
Tofindthereciprocalof
g.x/De
x
D
X1
nD0
xn
;
(4.5.27)
weagainlethD1in(4.5.25).If
.e
x
/
1
D
X1
nD0
a
n
x
n
;
then
1D
X1
nD0
a
n
x
n
X1
nD0
x
n
!
D
X1
nD0
c
n
x
n
;
where
c
n
D
Xn
rD0
a
r
.nr/Š
:
FromCorollary4.5.7,c
0
Da
0
D1andc
n
D0ifn1;hence,
a
n
D
n1
rD0
a
r
.nr/Š
; n1:
(4.5.28)
Solvingtheseequationssuccessivelyfora
0
,a
1
,...yields
a
1
D
1
(4.5.1)D1;
a
2
D
1
.1/C
1
.1/
D
1
2
;
a
3
D
1
.1/C
1
.1/C
1
1
2

D
1
6
a
4
D
1
.1/C
1
.1/C
1
1
2
C
1
1
6

D
1
24
:
Fromthis,weseethat
a
k
D
.1/k
for0k4andareledtoconjecturethatthisholdsforallk.Toprovethisbyinduction,
weassumethatitissofor0kn1andcomputefrom(4.5.28):
a
n
D
n1
X
rD0
1
.nr/Š
.1/
r
D
1
n1
rD0
.1/
r
n
r
!
(Exercise1.2.19
(a)
)
D
.1/n
(Exercise1.2.19
(b)
):
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Section4.5
PowerSeries
273
Thus,wehaveshownthat
.e
x
/
1
D
X1
nD0
.1/
n
x
n
:
Sincethisispreciselytheseriesthatresultsifx isreplacedbyxin(4.5.27), wehave
verifiedafundamentalpropertyoftheexponentialfunction:that
.e
x
/
1
De
x
:
ThisalsofollowsfromExample4.3.26.
Abel’sTheorem
FromTheorem4.5.4,weknowthatafunctionf definedbyaconvergentpowerseries
f.x/D
X1
nD0
a
n
.xx
0
/
n
; jxx
0
j<R;
(4.5.29)
iscontinuousintheopeninterval.x
0
R;x
0
CR/.Thenexttheoremconcernsthebehavior
off asxapproachesanendpointoftheintervalofconvergence.
Theorem4.5.12(Abel’sTheorem)
Letf bedefinedbyapowerseries(4.5.29)
withfiniteradiusofconvergenceR:
(a)
If
P
1
nD0
a
n
R
n
converges;then
lim
x!.x
0
CR/
f.x/D
X1
nD0
a
n
R
n
:
(b)
If
P
1
nD0
.1/
n
a
n
R
n
converges;then
lim
x!.x
0
R/C
f.x/D
X1
nD0
.1/
n
a
n
R
n
:
Proof
Weconsiderasimplerproblemfirst.Let
g.y/D
X1
nD0
b
n
y
n
and
X1
nD0
b
n
Ds (finite):
Wewillshowthat
lim
y!1
g.y/Ds:
(4.5.30)
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274 Chapter4
InfiniteSequencesandSeries
FromExample4.5.11,
g.y/D.1y/
1
X
nD0
s
n
y
n
;
(4.5.31)
where
s
n
Db
0
Cb
1
CCb
n
:
Since
1
1y
D
X1
nD0
y
n
andtherefore 1D.1y/
X1
nD0
y
n
; jyj<1;
(4.5.32)
wecanmultiplythroughbysandwrite
sD.1y/
X1
nD0
sy
n
; jyj<1:
Subtractingthisfrom(4.5.31)yields
g.y/sD.1y/
1
X
nD0
.s
n
s/y
n
; jyj<1:
If>0,chooseN sothat
js
n
sj< if nNC1:
Then,if0<y<1,
jg.y/sj.1y/
XN
nD0
js
n
sjy
n
C.1y/
X1
nDNC1
js
n
sjy
n
<.1y/
XN
nD0
js
n
sjy
n
C.1y/y
NC1
X1
nD0
y
n
<.1y/
XN
nD0
js
n
sjC;
becauseofthesecondequalityin(4.5.32).Therefore,
jg.y/sj<2
if
.1y/
XN
nD0
js
n
sj<:
Thisproves(4.5.30).
Toobtain
(a)
fromthis,letb
n
Da
n
R
n
andg.y/D f.x
0
CRy/;toobtain
(b)
,let
b
n
D.1/
n
a
n
R
n
andg.y/Df.x
0
Ry/.
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Section4.5
PowerSeries
275
Example4.5.16
Theseries
f.x/D
1
1Cx
D
1
X
nD0
.1/
n
x
n
divergesatx D1,whilelim
x!1
f.x/D 1=2. . ThisshowsthattheconverseofAbel’s
theoremisfalse.Integratingtheseriestermbytermyields
log.1Cx/D
X1
nD0
.1/
n
x
nC1
nC1
; jxj<1;
wherethepowerseriesconvergesatxD1,andAbel’stheoremimpliesthat
log2D
X1
nD0
.1/
nC1
nC1
:
Example4.5.17
Ifq0,thebinomialseries
X1
nD0
q
n
!
x
n
converges absolutelyforx x D ˙1. Thisisobviousifqisanonnegativeinteger, , andit
followsfromRaabe’stestforotherpositivevaluesofq,since
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
D
ˇ
ˇ
ˇ
ˇ
ˇ
q
nC1
!
q
n
!
ˇ
ˇ
ˇ
ˇ
ˇ
D
nq
nC1
; n>q;
and
lim
n!1
n
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
1
D lim
n!1
n
nq
nC1
1
D lim
n!1
n
nC1
.q1/Dq1:
Therefore,Abel’stheoremand(4.5.21)implythat
X1
nD0
q
n
!
D2
q
and
X1
nD0
.1/
n
q
n
!
D0; q0:
4.5Exercises
1.
ThepossibilitieslistedinTheorem4.5.2
(c)
forbehaviorofapowerseriesatthe
endpointsofitsintervalofconvergencedonotincludeabsoluteconvergenceatone
endpointandconditionalconvergenceordivergenceattheother. Whycan’tthese
occur?
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276 Chapter4
InfiniteSequencesandSeries
2.
Findtheradiusofconvergence.
(a)
X
nC1
n
n
2
Œ2C.1/
n
n
x
n
(b)
P
2
p
n
.x1/
n
(c)
X
2Csin
n
6
n
.xC2/
n
(d)
P
n
p
n
x
n
(e)
X
x
n
n
3. (a)
Prove:Iffa
n
r
n
gisboundedandjx
1
x
0
j<r,then
P
a
n
.x
1
x
0
/
n
con-
verges.
(b)
Prove: If
P
a
n
.xx
0
/hasradiusofconvergenceRandjx
1
x
0
j> R,
thenfa
n
.x
1
x
0
/ngisunbounded.
4.
Prove:Ifgisarationalfunctiondefinedforallnonnegativeintegers,then
P
a
n
x
n
and
P
a
n
g.n/x
n
havethesameradiusofconvergence.H
INT
:UseExercise4.1.30.a/:
5.
Supposethatf.x/D
P
a
n
.xx
0
/
n
hasradiusofconvergenceRand0<r <
R
1
<R.Showthatthereisanintegerksuchthat
ˇ
ˇ
ˇ
ˇ
ˇ
f.x/
Xk
nD0
a
n
.xx
0
/
n
ˇ
ˇ
ˇ
ˇ
ˇ
r
R
1
kC1
R
1
R
1
r
ifjxx
0
jrandkk.
6.
Supposethatkisapositiveintegerand
f.x/D
X1
nD0
a
n
x
n
hasradiusofconvergenceR.Showthattheseries
g.x/Df.x
k
/D
X1
nD0
a
n
x
kn
hasradiusofconvergenceR
1=k
.
7.
CompletetheproofofTheorem4.5.3byshowingthat
(a)
RD0iflim
n!1
ja
nC1
j
ı
ja
n
jD1;
(b)
RD1iflim
n!1
ja
nC1
j
ı
ja
n
jD0.
8.
Findtheradiusofconvergence.
(a)
P
.logn/x
n
(b)
P
2
n
n
p
.xC1/
n
(c)
X
.1/
n
2n
n
!
x
n
(d)
X
.1/
n
n
2
C1
n4n
.x1/
n
(e)
X
nn
.xC2/
n
(f)
X
˛.˛C1/.˛Cn1/
ˇ.ˇC1/.ˇCn1/
x
n
(˛,ˇ¤negativeinteger)
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Section4.5
PowerSeries
277
9.
Supposethata
n
¤0fornsufficientlylarge.Showthat
(a)
lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
 lim
n!1
ja
n
j
1=n
and
(b)
lim
n!1
ja
n
j
1=n
lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
:
ShowthatthisimpliesTheorem4.5.3.
10.
Giventhat
1
1x
D
X1
nD0
x
n
; jxj<1;
useTheorem4.5.4toexpress
P
1
nD0
n2xinclosedform.
11.
Thefunction
J
p
.x/D
X1
nD0
.1/
n
nŠ.nCp/Š
x
2
2nCp
.pD integer 0/
istheBesselfunctionoforderp.Showthat
(a)
J
0
0
DJ
1
.
(b)
J
0
p
D
1
2
.J
p1
J
pC1
/; p1.
(c)
x
2
J
00
p
CxJ
0
p
C.x
2
p
2
/J
p
D0.
12.
Giventhatthepowerseriesf.x/D
P
1
nD0
a
n
xsatisfies
f
0
.x/D2xf.x/; f.0/D1;
findfa
n
g.Doyourecognizef?
13.
Let
f.x/D
X1
nD0
a
n
x
n
; jxj<R;
andg.x/Df.x
k
/,wherekisapositiveinteger.Showthat
g
.r/
.0/D0 if r¤kn and g
.kn/
.0/D
.kn/Š
f
.n/
.0/; n0:
14.
Let
f.x/D
X1
nD0
a
n
.xx
0
/
n
; jxx
0
j<R;
andf.t
n
/ D D 0, , where t
n
¤ x
0
andlim
n!1
t
n
D x
0
. Showthatf.x/  0
.jxx
0
j<R/.H
INT
:Rolle’stheoremhelpshere:
15.
ProveTheorem4.5.8.
16.
Express
Z
x
1
logt
t1
dt
asapowerseriesinx1andfindtheradiusofconvergenceoftheseries.
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278 Chapter4
InfiniteSequencesandSeries
17.
Bysubstitutingx
2
forxinthegeometricseries,weobtain
1
1Cx2
D
X1
nD0
.1/
n
x
2n
; jxj<1:
Usethistoexpressf.x/ D Tan
1
x .f.0/ D D 0/asapowerseriesinx. Then
evaluateallderivativesoff atx
0
D0,andfindaseriesofconstantsthatconverges
to=6.
18.
Prove:If
f.x/D
X1
nD0
a
n
.xx
0
/
n
; jxx
0
j<R;
andFisanantiderivativeoffon.x
0
R;x
0
CR/,then
F.x/DCC
X1
nD0
a
n
nC1
.xx
0
/
nC1
; jxx
0
j<R;
whereCisaconstant.
19.
Supposethatsomederivativeoff canberepresentedbyapowerseriesinxx
0
inanintervalaboutx
0
.Showthatf andallitsderivativescanalso.
20.
VerifyEqn.(4.5.21)byshowingthat
.1Cx/
q
X1
nD0
q
n
!
x
n
D1; jxj<1;
H
INT
:Differentiate:
21.
ProveTheorem4.5.10.
22.
FindtheMaclaurinseriesofcoshxandsinhxfromthedefinitioninEqn.(4.5.16),
andalsobyapplyingTheorem4.5.10totheMaclaurinseriesfore
x
ande
x
.
23.
Giveanexamplewheretheradiusofconvergenceoftheproductoftwopowerseries
isgreaterthanthesmalleroftheradiiofconvergenceofthefactors.
24.
UseTheorem4.5.11tofindthefirstfournonzerotermsintheMaclaurin.
(a)
e
x
sinx
(b)
e
x
1Cx2
(c)
cosx
1Cx6
(d)
.sinx/log.1Cx/
25.
Derivetheidentity
2sinxcosxDsin2x
fromtheMaclaurinseriesforsinx,cosx,andsin2x.
26. (a)
Giventhat
.12xtCx
2
/
1=2
D
X1
nD0
P
n
.t/x
n
; jxj<1;
.A/
Section4.5
PowerSeries
279
if1<t<1,showthatP
0
.t/D1,P
1
.t/Dt,and
P
nC1
.t/D
2nC1
nC1
tP
n
.t/
n
nC1
P
n1
.t/; n1:
H
INT
:Firstdifferentiate(A)withrespecttox:
(b)
Showfrom
(a)
thatP
n
isapolynomialofdegreen. ItisthenthLegendre
polynomial,and.12xtCx
2
/
1=2
isthegeneratingfunctionofthesequence
fP
n
g.
27.
Define(ifnecessary)thegivenfunctionsoastobecontinuousatx
0
D0,andfind
thefirstfournonzerotermsofitsMaclaurinseries.
(a)
xe
x
sinx
(b)
cosx
1CxCx2
(c)
secx
(d)
xcscx
(e)
sin2x
sinx
28.
Leta
0
Da
1
D5anda
nC1
Da
n
6a
n1
;n1.
(a)
ExpressF.x/D
P
1
nD0
a
n
x
n
inclosedform.
(b)
WriteFasthedifferenceoftwogeometricseries,andfindanexplicitformula
fora
n
.
29.
StartingfromtheMaclaurinseries
log.1x/D
X1
nD0
x
nC1
nC1
; jxj<1;
useAbel’stheoremtoevaluate
X1
nD0
1
.nC1/.nC2/
:
30.
InExample4.5.17wesawthat
X1
nD0
q
n
!
D2
q
; q0:
Showthatthisalsoholdsfor1<q <0,butnotforq1. H
INT
:SeeExer-
cise4.1.35:
31. (a)
Prove:If
P
1
nD0
b
n
converges,thentheseriesg.x/D
P
1
nD0
b
n
x
n
converges
uniformlyonŒ0;1.H
INT
:If>0,thereisanintegerN suchthat
jb
n
Cb
nC1
CCb
m
j< if n;mN:
Usesummationbypartstoshowthatthen
jb
n
x
n
Cb
n1
x
n1
CCb
m
x
m
j<2 if 0x<1; n;mN:
ThisisalsoknownasAbel’stheorem:
280 Chapter4
InfiniteSequencesandSeries
(b)
Showthat
(a)
impliestherestrictedformofTheorem4.5.12(concerningg)
provedinthetext.
32.
UseExercise4.5.31toshowthatif
P
1
nD0
a
n
,
P
1
nD0
b
n
,andtheirCauchyproduct
P
1
nD0
c
n
allconverge,then
X1
nD0
a
n
X1
nD0
b
n
!
D
X1
nD0
c
n
:
33.
Prove:If
g.x/D
1
X
nD0
b
n
x
n
; jxj<1;
andb
n
0,then
X1
nD0
b
n
D lim
x!1
g.x/ (finiteorinfinite):
34.
Usethebinomialseriesandtherelation
d
dx
.sin
1
x/D.1x
2
/
1=2
toobtaintheMaclaurinseriesforsin
1
x .sin
1
0D0/. Deducefromthisseries
andExercise4.5.33that
X1
nD0
2n
n
!
1
22n.2nC1/
D
2
:
CHAPTER5
Real-ValuedFunctions
ofSeveralVariables
INTHISCHAPTERweconsiderreal-valuedfunctionofnvariables,wheren>1.
SECTION5.1dealswiththestructureofR
n
,thespaceoforderedn-tuplesofrealnumbers,
whichwecallvectors. Wedefinethesumoftwovectors, , theproductofavectoranda
realnumber, thelengthofavector, andtheinnerproductoftwovectors. Westudythe
arithmeticpropertiesofR
n
,includingSchwarz’sinequalityandthetriangleinequality.We
defineneighborhoodsandopensetsinR
n
,defineconvergenceofasequenceofpointsin
R
n
,andextendtheHeine–BoreltheoremtoR
n
. Thesectionconcludeswithadiscussion
ofconnectedsubsetsofR
n
.
SECTION5.2dealswithboundedness,limits,continuity,anduniformcontinuityofafunc-
tionofnvariables;thatis,afunctiondefinedonasubsetofR
n
.
SECTION5.3definesdirectionalandpartialderivativesofareal-valuedfunctionofn
variables.Thisisfollowedbythedefinitionofdifferentiablityofsuchfunctions.Wedefine
thedifferentialofsuchafunctionandgiveageometricinterpretationofdifferentiablity.
SECTION5.4dealswiththechainruleandTaylor’stheoremforareal-valuedfunctionof
nvariables.
5.1STRUCTUREOF
RRR
n
Inthischapterwestudyfunctionsdefinedonsubsetsoftherealn-dimensionalspaceR
n
,
whichconsistsofallorderedn-tuplesXD .x
1
;x
2
;:::;x
n
/ofrealnumbers, calledthe
coordinatesorcomponentsofX.ThisspaceissometimescalledEuclidean n-space.
InthissectionweintroduceanalgebraicstructureforR
n
.Wealsoconsideritstopologi-
calproperties;thatis,propertiesthatcanbedescribedintermsofaspecialclassofsubsets,
theneighborhoodsinR
n
.InSection1.3westudiedthetopologicalpropertiesofR
1
,which
wewillcontinuetodenotesimplyasR. MostofthedefinitionsandproofsinSection1.3
werestatedintermsofneighborhoodsinR. WewillseethattheycarryovertoR
n
ifthe
conceptofneighborhoodinR
n
issuitablydefined.
281
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