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262 Chapter4
InfiniteSequencesandSeries
Defining
b
n
D.nCk/.nCk1/.nC1/a
nCk
;
(4.5.12)
werewritethisas
f
.k/
.x/D
X1
nD0
b
n
.xx
0
/
n
; jxx
0
j<R:
ByTheorem4.5.4,wecandifferentiatethisseriestermbytermtoobtain
f
.kC1/
.x/D
X1
nD1
nb
n
.xx
0
/
n1
; jxx
0
j<R:
Substitutingfrom(4.5.12)forb
n
yields
f
.kC1/
.x/D
X1
nD1
.nCk/.nCk1/.nC1/na
nCk
.xx
0
/
n1
; jxx
0
j<R:
Shiftingthesummationindexyields
f
.kC1/
.x/D
X1
nDkC1
n.n1/.nk/a
n
.xx
0
/
nk1
; jxx
0
j<R;
whichis(4.5.11)withkreplacedbykC1.Thiscompletestheinduction.
Example4.5.6
InExample4.4.10wesawthat
1
1x
D
1
X
nD0
x
n
; jxj<1:
Repeateddifferentiationyields
.1x/kC1
D
X1
nDk
n.n1/.nkC1/x
nk
D
X1
nD0
.nCk/.nCk1/.nC1/x
n
; jxj<1;
so
1
.1x/kC1
D
X1
nD0
nCk
k
!
x
n
; jxj<1:
Example4.5.7
BythemethodofExample4.5.5,itcanbeshownthattheseries
S.x/D
1
X
nD0
.1/
n
x
2nC1
.2nC1/Š
and
C.x/D
1
X
nD0
.1/
n
x
2n
.2n/Š
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Section4.5
PowerSeries
263
convergeforallx.Differentiatingyields
S
0
.x/D
1
X
nD0
.1/
n
x
n
.2n/Š
DC.x/
and
C
0
.x/D
X1
nD1
.1/
n
x
2n1
.2n1/Š
D
X1
nD0
.1/
n
x
2nC1
.2nC1/Š
DS.x/:
Theseresultsshouldnotsurpriseyouifyourecallthat
S.x/Dsinx and C.x/Dcosx:
(Wewillsoonprovethis.)
Theorem4.5.5hastwoimportantcorollaries.
Corollary4.5.6
If
f.x/D
X1
nD0
a
n
.xx
0
/
n
; jxx
0
j<R;
then
a
n
D
f.n/.x
0
/
:
Proof
SettingxDx
0
in(4.5.11)yields
f
.k/
.x
0
/DkŠa
k
:
Corollary4.5.7(UniquenessofPowerSeries)
If
X1
nD0
a
n
.xx
0
/
n
D
X1
nD0
b
n
.xx
0
/
n
(4.5.13)
forallxinsomeinterval.x
0
r;x
0
Cr/;then
a
n
Db
n
; n0:
(4.5.14)
Proof
Let
f.x/D
X1
nD0
a
n
.xx
0
/
n
and g.x/D
X1
nD0
b
n
.xx
0
/
n
:
FromCorollary4.5.6,
a
n
D
f
.n/
.x
0
/
and b
n
D
g
.n/
.x
0
/
:
(4.5.15)
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264 Chapter4
InfiniteSequencesandSeries
From(4.5.13),f Dgin.x
0
r;x
0
Cr/.Therefore,
f
.n/
.x
0
/Dg
.n/
.x
0
/; n0:
Thisand(4.5.15)imply(4.5.14).
Theorems4.4.19and4.5.2implythefollowingtheorem. Weleavetheprooftoyou
(Exercise4.5.15).
Theorem4.5.8
Ifx
1
andx
2
areintheintervalofconvergenceof
f.x/D
X1
nD0
a
n
.xx
0
/
n
;
then
Z
x
2
x
1
f.x/dxD
X1
nD0
a
n
nC1
.x
2
x
0
/
nC1
.x
1
x
0
/
nC1
I
thatis;apowerseriesmaybeintegratedtermbytermbetweenanytwopointsinitsinterval
ofconvergence:
Example4.5.16presentsanapplicationofthistheorem.
Taylor’sSeries
Sofarwehaveaskedforwhatvaluesofxagivenpowerseriesconverges,andwhatare
thepropertiesofitssum.Nowweaskarelatedquestion:Whatpropertiesguaranteethata
givenfunctionf canberepresentedasthesumofaconvergentpowerseriesinxx
0
?A
partialanswertothisquestionisprovidedbywhatwealreadyknow:Theorem4.5.5tellsus
thatf musthavederivativesofallordersinsomeneighborhoodofx
0
,andCorollary4.5.6
tellsusthattheonlypowerseries inxx
0
thatcanpossiblyconvergetof insucha
neighborhoodis
X1
nD0
f
.n/
.x
0
/
.xx
0
/
n
:
(4.5.16)
ThisiscalledtheTaylorseriesoff aboutx
0
(also,theMaclaurinseriesoff,ifx
0
D0).
Themthpartialsumof(4.5.16)istheTaylorpolynomial
T
m
.x/D
Xm
nD0
f
.n/
.x
0
/
.xx
0
/
n
;
definedinSection2.5.
TheTaylorseriesofaninfinitelydifferentiablefunctionf mayconvergetoasumdif-
ferentfromf.Forexample,thefunction
f.x/D
e
1=x
2
; x¤0;
0;
xD0;
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Section4.5
PowerSeries
265
isinfinitelydifferentiableon.1;1/andf
.n/
.0/D0forn0(Exercise2.5.1),soits
Maclaurinseriesisidenticallyzero.
TheanswertoourquestionisprovidedbyTaylor’stheorem(Theorem 2.5.4), which
saysthatiff isinfinitelydifferentiableon.a;b/andxandx
0
arein.a;b/then,forevery
integern0,
f.x/T
n
.x/D
f.nC1/.c
n
/
.nC1/Š
.xx
0
/
n1
;
(4.5.17)
wherec
n
isbetweenxandx
0
.Therefore,
f.x/D
X1
nD0
f
.n/
.x
0
/
.xx
0
/
n
foranxin.a;b/ifandonlyif
lim
n!1
f
.nC1/
.c
n
/
.nC1/Š
.xx
0
/
nC1
D0:
Itisnotalwayseasytocheckthiscondition,becausethesequencefc
n
gisusuallynotpre-
ciselyknown,orevenuniquelydefined;however,thenexttheoremissufficientlygeneral
tobeuseful.
Theorem4.5.9
Supposethatf isinfinitelydifferentiableonanintervalIand
lim
n!1
r
n
kf
.n/
k
I
D0:
(4.5.18)
Then;ifx
0
2I
0
;theTaylorseries
1
X
nD0
f
.n/
.x
0
/
.xx
0
/
n
convergesuniformlytof on
I
r
DI \Œx
0
r;x
0
Cr:
Proof
From(4.5.17),
kf T
n
k
I
r
r
nC1
.nC1/Š
kf
.nC1/
k
I
r
r
nC1
.nC1/Š
kf
.nC1/
k
I
;
so(4.5.18)impliestheconclusion.
Example4.5.8
Iff.x/Dsinx,thenkf
.k/
k
.1;1/
D1;k0.Since
lim
n!1
r
n
D0; 0<r<1
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266 Chapter4
InfiniteSequencesandSeries
(Example4.1.12),(4.5.18)holdsforallr.Since
f
.2m/
.0/D0 and f
.2mC1/
.0/D.1/
m
; m0;
weseefromTheorem4.5.9,withID.1;1/,x
0
D0,andrarbitrary,that
sinxD
X1
nD0
.1/
n
x
2nC1
.2nC1/Š
; 1<x<1;
andtheconvergenceisuniformonboundedsets.
Asimilarargumentshowsthat
cosxD
X1
nD0
.1/
n
x
2n
.2n/Š
; 1<x<1;
withuniformconvergenceonboundedsets.
Example4.5.9
Iff.x/ D D e
x
, thenf
.k/
.x/ D D e
x
andkf
.k/
k
I
D e
r
, k k  0, , if
I DŒr;r.Since
lim
n!1
r
n
e
r
D0;
weconcludeasinExample4.5.8that
e
x
D
X1
nD0
x
n
; 1<x<1;
withuniformconvergenceonboundedsets.
Example4.5.10
Iff.x/D.1Cx/
q
,then
f
.n/
.x/
D
q
n
!
.1Cx/
qn
; so
f
.n/
.0/
D
q
n
!
(4.5.19)
(Example2.5.3).TheMaclaurinseries
1
X
nD0
q
n
!
x
n
iscalledthebinomialseries. WesawinExample2.5.3thatthisseriesequals.1Cx/for
allxifqisanonnegativeinteger. Wewillnowshowthatifqisanarbitraryrealnumber,
then
X1
nD0
q
n
!
x
n
Df.x/D.1Cx/
q
; 0x<1:
(4.5.20)
Since
Section4.5
PowerSeries
267
lim
n!1
ˇ
ˇ
ˇ
ˇ
ˇ
q
nC1
!
q
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
ˇ
qn
nC1
ˇ
ˇ
ˇ
ˇ
D1;
theradiusofconvergenceoftheseriesin(4.5.20)is1.From(4.5.19),
kf
.n/
k
Œ0;1
Œmax.1;2
q
/
ˇ
ˇ
ˇ
ˇ
ˇ
q
n
ˇ
ˇ
ˇ
ˇ
; n0:
Therefore,if0<r<1,
lim
n!1
rn
kf
.n/
k
Œ0;1
Œmax.1;2
q
/ lim
n!1
ˇ
ˇ
ˇ
ˇ
ˇ
q
n
!
ˇ
ˇ
ˇ
ˇ
ˇ
r
n
D0;
wherethelastequalityfollowsfromtheabsoluteconvergenceoftheseriesin(4.5.20)on
.1;1/.NowTheorem4.5.9implies(4.5.20).
Wecannotproveinthiswaythatthebinomialseriesconvergesto.1Cx/
q
on.1;0/.
ThisrequiresaformoftheremainderinTaylor’stheoremthatwehavenotconsidered,or
adifferentkindofproofaltogether(Exercise4.5.20).Thecompleteresultisthat
.1Cx/
q
D
X1
nD0
q
n
!
x
n
; 1<x<1;
(4.5.21)
forallq,and,aswesaidearlier,theidentityholdsforallxifqisanonnegativeinteger.
ArithmeticOperationswithPowerSeries
Wenowconsideradditionandmultiplicationofpowerseries,anddivisionofonebyan-
other.
Weleavetheproofofthenexttheoremtoyou(Exercise4.5.21).
Theorem4.5.10
If
f.x/D
1
X
nD0
a
n
.xx
0
/
n
; jxx
0
j<R
1
;
(4.5.22)
g.x/D
X1
nD0
b
n
.xx
0
/
n
; jxx
0
j<R
2
;
(4.5.23)
and˛andˇareconstants;then
˛f.x/Cˇg.x/D
X1
nD0
.˛a
n
Cˇb
n
/.xx
0
/
n
; jxx
0
j<R;
whereRminfR
1
;R
2
g:
268 Chapter4
InfiniteSequencesandSeries
Theorem4.5.11
Iff andgaregivenby(4.5.22)and(4.5.23);then
f.x/g.x/D
1
X
nD0
c
n
.xx
0
/
n
; jxx
0
j<R;
(4.5.24)
where
c
n
D
Xn
rD0
a
r
b
nr
D
Xn
rD0
a
nr
b
r
andRminfR
1
;R
2
g:
Proof
SupposethatR
1
 R
2
. Sincetheseries(4.5.22)and(4.5.23)convergeabso-
lutelytof.x/andg.x/ifjxx
0
j<R
1
,theirCauchyproductconvergestof.x/g.x/if
jxx
0
j<R
1
,byTheorem4.3.29.Thenthtermofthisproductis
Xn
rD0
a
r
.xx
0
/
r
b
nr
.xx
0
/
nr
D
Xn
rD0
a
r
b
nr
!
.xx
0
/
n
Dc
n
.xx
0
/
n
:
Example4.5.11
If
f.x/D
1
1x
D
X1
nD0
x
n
; jxj<1;
and
g.x/D
X1
nD0
b
n
x
n
; jxj<R;
then
g.x/
1x
D
X1
nD0
s
n
x
n
; jxj<minf1;Rg;
where
s
n
D.1/b
0
C.1/b
1
CC.1/b
n
Db
0
Cb
1
CCb
n
:
Example4.5.12
FromtheparagraphfollowingExample4.5.10,
.1Cx/
p
D
X1
nD0
p
n
!
x
n
; jxj<1;
and
.1Cx/
q
D
X1
nD0
q
n
!
x
n
; jxj<1:
Section4.5
PowerSeries
269
Since
.1Cx/
p
.1Cx/
q
D.1Cx/
pCq
D
X1
nD0
pCq
n
!
x
n
;
whiletheCauchyproductis
P
1
nD0
c
n
x
n
,with
c
n
D
Xn
rD0
p
r
q
nr
!
;
Corollary4.5.7impliesthat
c
n
D
pCq
n
!
:
Thisyieldstheidentity
pCq
n
!
D
n
X
rD0
p
r
q
nr
!
;
validforallpandq.
Thequotient
f.x/D
h.x/
g.x/
(4.5.25)
oftwopowerseries
h.x/D
X1
nD0
c
n
.xx
0
/
n
; jxx
0
j<R
1
;
and
g.x/D
X1
nD0
b
n
.xx
0
/
n
; jxx
0
j<R
2
;
canberepresentedasapowerseries
f.x/D
X1
nD0
a
n
.xx
0
/
n
(4.5.26)
withapositiveradiusofconvergence,providedthat
b
0
Dg.x
0
/¤0:
Thisissurelyplausible. Sinceg.x
0
/¤0andgiscontinuousnearx
0
,thedenominator
of(4.5.25)differsfromzeroonanintervalaboutx
0
. Therefore,f hasderivativesofall
ordersonthisinterval,becausegandhdo.However,theproofthattheTaylorseriesoff
aboutx
0
convergestof nearx
0
requirestheuseofthetheoryoffunctionsofacomplex
variable.Therefore,weomitit.However,itisstraightforwardtocomputethecoefficients
in(4.5.26)ifweacceptthevalidityoftheexpansion.Since
f.x/g.x/Dh.x/;
270 Chapter4
InfiniteSequencesandSeries
Theorem4.5.11impliesthat
Xn
rD0
a
r
b
nr
Dc
n
; n0:
Solvingtheseequationssuccessivelyyields
a
0
D
c
0
b
0
;
a
n
D
1
b
0
c
n
n1
rD0
b
nr
a
r
!
; n1:
Itisnotworthwhiletomemorizetheseformulas.Rather,itisusuallybettertoviewthe
procedureasfollows:Multiplytheseriesf (withunknowncoefficients)andgaccording
totheprocedureofTheorem4.5.11,equatetheresultingcoefficientswiththoseofh,and
solvetheresultingequationssuccessivelyfora
0
,a
1
,....
Example4.5.13
SupposethatwewishtofindthecoefficientsintheMaclaurinseries
tanxDa
0
Ca
1
xCa
2
x
2
C:
Wefirstobservethatsincetanxisanoddfunction,itsderivativesofevenordervanishat
x
0
D0,soa
2m
D0,m0.Therefore,
tanxDa
1
xCa
3
x
3
Ca
5
x
5
C:
Since
tanxD
sinx
cosx
;
itfollowsfromExample4.5.8that
a
1
xCa
3
x
3
Ca
5
x
5
CD
x
x3
6
C
x5
120
C
1
x2
2
C
x4
24
C
so
.a
1
xCa
3
x
3
Ca
5
x
5
C/
1
x
2
2
C
x
4
24
C
Dx
x
3
6
C
x
5
120
C;
or,accordingtoTheorem4.5.11,
a
1
xC
a
3
a
1
2
x
3
C
a
5
a
3
2
C
a
1
24
x
5
CDx
x
3
6
C
x
5
120
C:
FromCorollary4.5.7,coefficientsoflikepowersofxonthetwosidesofthisequation
mustbeequal;hence,
a
1
D1;
a
3
a
1
2
D
1
6
;
a
5
a
3
2
C
a
1
24
D
1
120
;
so
a
1
D1; a
3
D
1
6
C
1
2
.1/D
1
3
; a
5
D
1
120
C
1
2
1
3
1
24
.1/D
2
15
:
Section4.5
PowerSeries
271
Therefore,
tanxDxC
x
3
3
C
2
15
x
5
C:
Example4.5.14
Tofindthereciprocalofthepowerseries
g.x/D1Ce
x
D2C
X1
nD1
x
n
;
welethD1in(4.5.25).If
1
g.x/
D
X1
nD0
a
n
x
n
;
then
1D.a
0
Ca
1
xCa
2
x
2
Ca
3
x
3
C/
2CxC
x
2
2
C
x
3
6
C
D2a
0
C.a
0
C2a
1
/xC
a
0
2
Ca
1
C2a
2
x
2
C
a
0
6
C
a
1
2
Ca
2
C2a
3
x
3
C:
FromCorollary4.5.7,
2a
0
D1;
a
0
C2a
1
D0;
a
0
2
Ca
1
C2a
2
D0;
a
0
6
C
a
1
2
Ca
2
C2a
3
D0:
Solvingtheseequationssuccessivelyyields
a
0
D
1
2
;
a
1
D
a
0
2
D
1
4
;
a
2
D
1
2
a
0
2
Ca
1
D
1
2
1
4
1
4
D0;
a
3
D
1
2
a
0
6
C
a
1
2
Ca
2
D
1
2
1
12
1
8
C0
D
1
48
;
so
1
1Cex
D
1
2
x
4
C
x
3
48
C:
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