212 Chapter4
InfiniteSequencesandSeries
Thefollowingcorollaryoftheratiotestisthefamiliarratiorestfromcalculus.
Corollary4.3.15
Supposethata
n
>0.nk/and
lim
n!1
a
nC1
a
n
DL:
Then
(a)
P
a
n
<1ifL<1:
(b)
P
a
n
D1ifL>1:
ThetestisinconclusiveifLD1:
Example4.3.12
Theseries
P
a
n
D
P
nr
n1
convergesif0<r<1ordivergesif
r>1,since
a
nC1
a
n
D
.nC1/r
n
nrn1
D
1C
1
n
r;
so
lim
n!1
a
nC1
a
n
Dr:
Corollary4.3.15isinconclusiveifrD1,butthenCorollary4.3.6impliesthattheseries
diverges.
Theratiotestdoesnotimplythat
P
a
n
<1ifmerely
a
nC1
a
n
<1
(4.3.14)
forlargen,sincethiscouldoccurwithlim
n!1
a
nC1
=a
n
D1,inwhichcasethetestis
inconclusive.However,thenexttheoremshowsthat
P
a
n
<1if(4.3.14)isreplacedby
thestrongerconditionthat
a
nC1
a
n
1
p
n
forsomep>1andlargen.Italsoshowsthat
P
a
n
D1if
a
nC1
a
n
1
q
n
forsomeq<1andlargen.
Theorem4.3.16(Raabe’sTest)
Supposethata
n
>0forlargen:Let
MD
lim
n!1
n
a
nC1
a
n
1
and mD lim
n!1
n
a
nC1
a
n
1
:
Then
(a)
P
a
n
<1ifM<1:
(b)
P
a
n
D1ifm>1:
Thetestisinconclusiveifm1M:
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Section4.3
InfiniteSeriesofConstants
213
Proof (a)
Weneedtheinequality
1
.1Cx/p
>1px; x>0;p>0:
(4.3.15)
ThisfollowsfromTaylor’stheorem(Theorem2.5.4),whichimpliesthat
1
.1Cx/p
D1pxC
1
2
p.pC1/
.1Cc/pC2
x
2
;
where0<c<x.(Verify.)Sincethelasttermispositiveifp>0,thisimplies(4.3.15).
NowsupposethatM <p<1.Thenthereisanintegerksuchthat
n
a
nC1
a
n
1
<p; nk;
so
a
nC1
a
n
<1
p
n
; nk:
Hence,
a
nC1
a
n
<
1
.1C1=n/p
; nk;
ascanbeseenbylettingxD1=nin(4.3.15).Fromthis,
a
nC1
a
n
<
1
.nC1/p
1
np
; nk:
Since
P
1=n<1ifp>1,Theorem4.3.13
(a)
impliesthat
P
a
n
<1.
(b)
Hereweneedtheinequality
.1x/
q
<1qx; 0<x<1; 0<q<1:
(4.3.16)
ThisalsofollowsfromTaylor’stheorem,whichimpliesthat
.1x/
q
D1qxCq.q1/.1c/
q2
x
2
2
;
where0<c<x.
Nowsupposethat1<q<m.Thenthereisanintegerksuchthat
n
a
nC1
a
n
1
>q; nk;
so
a
nC1
a
n
1
q
n
; nk:
Ifq0,then
P
a
n
D1,byCorollary4.3.6.Hence,wemayassumethat0<q<1,so
thelastinequalityimpliesthat
a
nC1
a
n
>
1
1
n
q
; nk;
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214 Chapter4
InfiniteSequencesandSeries
ascanbeseenbysettingxD1=nin(4.3.16).Hence,
a
nC1
a
n
>
1
nq
1
.n1/q
; nk:
Since
P
1=n
q
D1ifq<1,Theorem4.3.13
(b)
impliesthat
P
a
n
D1.
Example4.3.13
If
X
a
n
D
X
˛.˛C1/.˛C2/.˛Cn1/
; ˛>0;
then
lim
n!1
a
nC1
a
n
D lim
n!1
nC1
˛Cn
D1;
sotheratiotestisinconclusive.However,
lim
n!1
n
a
nC1
a
n
1
D lim
n!1
n
nC1
˛Cn
1
D lim
n!1
n.1˛/
˛Cn
D1˛;
soRaabe’stestimpliesthat
P
a
n
<1if˛>2and
P
a
n
D1if0<˛<2. Raabe’s
testisinconclusiveif˛D2,butthentheseriesbecomes
X
.nC1/Š
D
X
1
nC1
;
whichweknowisdivergent.
Example4.3.14
Considertheseries
P
a
n
,where
a
2m
D
.mŠ/
2
˛.˛C1/.˛Cm/ˇ.ˇC1/.ˇCm/
and
a
2mC1
D
.mŠ/
2
.mC1/
˛.˛C1/.˛Cm/ˇ.ˇC1/.ˇCmC1/
;
with0<˛<ˇ.Since
2m
a
2mC1
a
2m
1
D2m
mC1
ˇCmC1
1
D
2mˇ
ˇCmC1
and
.2mC1/
a
2mC2
a
2mC1
1
D.2mC1/
mC1
˛CmC1
1
D
.2mC1/˛
˛CmC1
;
wehave
lim
n!1
n
a
nC1
a
n
1
D2˛ and
lim
n!1
n
a
nC1
a
n
1
D2ˇ:
Raabe’stestimpliesthat
P
a
n
<1if˛>1=2and
P
a
n
D1ifˇ<1=2.Thetestis
inconclusiveif0<˛1=2ˇ.
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Section4.3
InfiniteSeriesofConstants
215
Thenexttheorem,whichwillbeusefulwhenwestudypowerseries(Section4.5),con-
cludesourdiscussionofserieswithnonnegativeterms.
Theorem4.3.17(Cauchy’sRoot Test)
Ifa
n
0fornk;then
(a)
P
a
n
<1if
lim
n!1
a
1=n
n
<1:
(b)
P
a
n
D1if
lim
n!1
a
1=n
n
>1:
Thetestisinconclusiveif
lim
n!1
a
1=n
n
D1:
Proof (a)
If
lim
n!1
a
1=n
n
<1,thereisanrsuchthat0<r <1anda
1=n
n
<rfor
largen.Therefore,a
n
<r
n
forlargen.Since
P
r
n
<1,thecomparisontestimpliesthat
P
a
n
<1.
(b)
If
lim
n!1
a
1=n
n
>1,thena
1=n
n
>1forinfinitelymanyvaluesofn,so
P
a
n
D1,
byCorollary4.3.6.
Example4.3.15
Cauchy’sroottestisinconclusiveif
X
a
n
D
X
1
np
;
becausethen
lim
n!1
a
1=n
n
D lim
n!1
1
np
1=n
D lim
n!1
exp
p
n
logn
D1
forallp. However, , weknowfrom theintegraltestthat
P
1=n
p
< 1ifp > 1and
P
1=n
p
D1ifp1.
Example4.3.16
If
X
a
n
D
X
2Csin
n
4
n
r
n
;
then
lim
n!1
a
1=n
n
D
lim
n!1
2Csin
n
4
rD3r;
andso
P
a
n
<1ifr < < 1=3and
P
a
n
D 1ifr r > > 1=3. Thetestisinconclusiveif
rD1=3,butthenja
8mC2
jD1form0,so
P
a
n
D1,byCorollary4.3.6.
AbsoluteandConditional Convergence
Wenowdroptheassumptionthatthetermsof
P
a
n
arenonnegativeforlargen. Inthis
case,
P
a
n
mayconvergeintwoquitedifferentways.Thefirstisdefinedasfollows.
Definition4.3.18
Aseries
P
a
n
convergesabsolutely,orisabsolutelyconvergent;if
P
ja
n
j<1:
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216 Chapter4
InfiniteSequencesandSeries
Example4.3.17
Aconvergentseries
P
a
n
ofnonnegativetermsisabsolutelyconver-
gent,since
P
a
n
and
P
ja
n
jarethesame. Moregenerally,anyconvergentserieswhose
termsareofthesamesignforsufficientlylargenconvergesabsolutely(Exercise4.3.22).
Example4.3.18
Considertheseries
X
sinn
np
;
(4.3.17)
whereisarbitraryandp>1.Since
ˇ
ˇ
ˇ
ˇ
sinn
np
ˇ
ˇ
ˇ
ˇ
1
np
and
P
1=n
p
<1ifp>1,thecomparisontestimpliesthat
X
ˇ
ˇ
ˇ
ˇ
sinn
np
ˇ
ˇ
ˇ
ˇ
<1; p>1:
Therefore,(4.3.17)convergesabsolutelyifp>1.
Example4.3.19
If0<p<1,thentheseries
X
.1/
n
np
doesnotconvergeabsolutely,since
X
ˇ
ˇ
ˇ
ˇ
.1/
n
np
ˇ
ˇ
ˇ
ˇ
D
X
1
np
D1:
However,theseriesconverges,bythealternatingseriestest,whichweprovebelow.
Anytestforconvergenceofaserieswithnonnegativetermscanbeusedtotestanarbi-
traryseries
P
a
n
forabsoluteconvergencebyapplyingitto
P
ja
n
j.Weusedthecompar-
isontestthiswayinExamples4.3.18and4.3.19.
Example4.3.20
Totesttheseries
X
a
n
D
X
.1/
n
˛.˛C1/.˛Cn1/
; ˛>0;
forabsoluteconvergence,weapplyRaabe’stestto
X
a
n
D
X
˛.˛C1/.˛Cn1/
:
FromExample4.3.13,
P
ja
n
j<1if˛>2and
P
ja
n
jD1if˛<2.Therefore,
P
a
n
convergesabsolutelyif˛>2,butnotif˛<2.Noticethatthisdoesnotimplythat
P
a
n
divergesif˛<2.
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Section4.3
InfiniteSeriesofConstants
217
TheproofofthenexttheoremisanalogoustotheproofofTheorem3.4.9.Weleaveitto
you(Exercise4.3.24).
Theorem4.3.19
If
P
a
n
convergesabsolutely;then
P
a
n
converges:
Forexample,Theorem4.3.19impliesthat
X
sinn
np
convergesifp>1,sinceitthenconvergesabsolutely(Example4.3.18).
TheconverseofTheorem4.3.19isfalse;aseriesmayconvergewithoutconvergingabso-
lutely.Wesaythenthattheseriesconvergesconditionally,orisconditionallyconvergent;
thus,
P
.1/n=nconvergesconditionallyif0 < p  1.
Dirichlet’sTestforSeries
ExceptforTheorem4.3.5andCorollary4.3.6,theconvergencetestswehavestudiedso
farapplyonlytoserieswhosetermshavethesamesignforlargen. Thefollowingtheo-
remdoesnotrequirethis.ItisanalogoustoDirichlet’stestforimproperintegrals(Theo-
rem3.4.10).
Theorem4.3.20(Dirichlet’sTestforSeries)
Theseries
P
1
nDk
a
n
b
n
con-
vergesiflim
n!1
a
n
D0;
X
ja
nC1
a
n
j<1;
(4.3.18)
and
jb
k
Cb
kC1
CCb
n
jM; nk;
(4.3.19)
forsomeconstantM:
Proof
TheproofissimilartotheproofofDirichlet’stestforintegrals.Define
B
n
Db
k
Cb
kC1
CCb
n
; nk
andconsiderthepartialsumsof
P
1
nDk
a
n
b
n
:
S
n
Da
k
b
k
Ca
kC1
b
kC1
CCa
n
b
n
; nk:
(4.3.20)
Bysubstituting
b
k
DB
k
and b
n
DB
n
B
n1
; nkC1;
into(4.3.20),weobtain
S
n
Da
k
B
k
Ca
kC1
.B
kC1
B
k
/CCa
n
.B
n
B
n1
/;
whichwerewriteas
S
n
D.a
k
a
kC1
/B
k
C.a
kC1
a
kC2
/B
kC1
C
C.a
n1
a
n
/B
n1
Ca
n
B
n
:
(4.3.21)
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218 Chapter4
InfiniteSequencesandSeries
(Theprocedurethatledfrom(4.3.20)to(4.3.21)iscalledsummationbyparts.Itisanalo-
goustointegrationbyparts.)Now(4.3.21)canbeviewedas
S
n
DT
n1
Ca
n
B
n
;
(4.3.22)
where
T
n1
D.a
k
a
kC1
/B
k
C.a
kC1
a
kC2
/B
kC1
CC.a
n1
a
n
/B
n1
I
thatis,fT
n
gisthesequenceofpartialsumsoftheseries
X1
jDk
.a
j
a
jC1
/B
j
:
(4.3.23)
Since
j.a
j
a
jC1
/B
j
jMja
j
a
jC1
j
from(4.3.19),thecomparisontestand(4.3.18)implythattheseries(4.3.23)converges
absolutely.Theorem4.3.19nowimpliesthatfT
n
gconverges.LetT Dlim
n!1
T
n
.Since
fB
n
gisboundedandlim
n!1
a
n
D0,weinferfrom(4.3.22)that
lim
n!1
S
n
D lim
n!1
T
n1
C lim
n!1
a
n
B
n
DTC0DT:
Therefore,
P
a
n
b
n
converges.
Example4.3.21
ToapplyDirichlet’stestto
X1
nD2
sinn
nC.1/n
; ¤k (kDinteger);
wetake
a
n
D
1
nC.1/n
and b
n
Dsinn:
Thenlim
n!1
a
n
D0,and
ja
nC1
a
n
j<
3
n.n1/
(verify),so
X
ja
nC1
a
n
j<1:
Now
B
n
Dsin2Csin3CCsinn:
ToshowthatfB
n
gisbounded,weusethetrigonometricidentity
sinrD
cos
r
1
2
cos
rC
1
2
2sin.=2/
; ¤2k;
Section4.3
InfiniteSeriesofConstants
219
towrite
B
n
D
.cos
3
2
cos
5
2
/C.cos
5
2
cos
7
2
/CC
cos
n
1
2
cos.nC
1
2
/
2sin.=2/
D
cos
3
2
cos.nC
1
2
/
2sin.=2/
;
whichimpliesthat
jB
n
j
ˇ
ˇ
ˇ
ˇ
1
sin.=2/
ˇ
ˇ
ˇ
ˇ
; n2:
Sincefa
n
gandfb
n
gsatisfythehypothesesofDirichlet’stheorem,
P
a
n
b
n
converges.
Dirichlet’stesttakesasimplerformiffa
n
gisnonincreasing,asfollows.
Corollary4.3.21(Abel’sTest)
Theseries
P
a
n
b
n
convergesifa
nC1
a
n
for
nk;lim
n!1
a
n
D0;and
jb
k
Cb
kC1
CCb
n
jM; nk;
forsomeconstantM:
Proof
Ifa
nC1
a
n
,then
Xm
nDk
ja
nC1
a
n
jD
Xm
nDk
.a
n
a
nC1
/Da
k
a
mC1
:
Sincelim
m!1
a
mC1
D0,itfollowsthat
X1
nDk
ja
nC1
a
n
jDa
k
<1:
Therefore,thehypothesesofDirichlet’stestaresatisfied,so
P
a
n
b
n
converges.
Example4.3.22
Theseries
X
sinn
np
;
whichweknowisconvergentifp > 1(Example4.3.18),alsoconvergesif0< p 1.
ThisfollowsfromAbel’stest,witha
n
D1=n
p
andb
n
Dsinn(seeExample4.3.21).
ThealternatingseriestestfromcalculusfollowseasilyfromAbel’stest.
Corollary4.3.22(AlternatingSeriesTest)
Theseries
P
.1/
n
a
n
converges
if0a
nC1
a
n
andlim
n!1
a
n
D0:
220 Chapter4
InfiniteSequencesandSeries
Proof
Letb
n
D .1/
n
; thenfjB
n
jg is asequenceofzeros andones andtherefore
bounded.TheconclusionnowfollowsfromAbel’stest.
GroupingTermsinaSeries
Thetermsofafinitesumcanbegroupedbyinsertingparenthesesarbitrarily.Forexample,
.1C7/C.6C5/C4D.1C7C6/C.5C4/D.1C7/C.6C5C4/:
Accordingtothe nexttheorem, thesame istrueofaninfiniteseries thatconverges or
divergesto˙1.
Theorem4.3.23
Supposethat
P
1
nDk
a
n
DA;where1A1:Letfn
j
g
1
1
be
anincreasingsequenceofintegers,withn
1
k.Define
b
1
Da
k
CCa
n
1
;
b
2
Da
n
1
C1
CCa
n
2
;
:
:
:
b
r
Da
n
r1
C1
CCa
n
r
:
Then
X1
jD1
b
n
j
DA:
Proof
IfT
r
istherthpartialsumof
P
1
jD1
b
n
j
andfA
n
gisthenthpartialsum of
P
1
sDk
a
s
,then
T
r
Db
1
Cb
2
CCb
r
D.a
1
CCa
n
1
/C.a
n
1
C1
CCa
n
2
/CC.a
n
r1
C1
CCa
n
r
/
DA
n
r
:
Thus,fT
r
gisasubsequenceoffA
n
g,solim
r!1
T
r
Dlim
n!1
A
n
DAbyTheorem4.2.2.
Example4.3.23
If
P
1
nD0
.1/
n
a
n
satisfies thehypothesesofthealternatingseries
testandconvergestothesumS,Theorem4.3.23enablesustowrite
SD
Xk
nD0
.1/
n
a
n
C.1/
kC1
X1
jD1
.a
kC2j1
a
kC2j
/
and
SD
Xk
nD0
.1/
n
a
n
C.1/
kC1
2
4
a
kC1
X1
jD1
.a
kC2j
a
kC2j1
/
3
5
:
Since0a
nC1
a
n
,thesetwoequationsimplythatSS
k
isbetween0and.1/
k1
a
kC1
.
Section4.3
InfiniteSeriesofConstants
221
Example4.3.24
Introducingparenthesesinsomedivergentseries canyieldseem-
inglycontradictoryresults.Forexample,itistemptingtowrite
X1
nD1
.1/
nC1
D.11/C.11/CD0C0C
andconcludethat
P
1
nD1
.1/
n
D0,butequallytemptingtowrite
X1
nD1
.1/
nC1
D1.11/.11/
D100
andconcludethat
P
1
nD1
.1/
nC1
D 1. Ofcourse,thereisnocontradictionhere, , since
Theorem4.3.23doesnotapplytothisseries,andneitheroftheseoperationsislegitimate.
RearrangementofSeries
Afinitesumisnotchangedbyrearrangingitsterms;thus,
1C3C7D1C7C3D3C1C7D3C7C1D7C1C3D7C3C1:
Thisisnottrueofallinfiniteseries. Letussaythat
P
b
n
isarearrangementof
P
a
n
if
thetwoserieshavethesameterms,writteninpossiblydifferentorders. Sincethepartial
sumsofthetwoseriesmayformentirelydifferentsequences,thereisnoapparentreason
toexpectthemtoexhibitthesameconvergenceproperties,andingeneraltheydonot.
Weareinterestedinwhathappensifwerearrangethetermsofaconvergentseries. We
willseethateveryrearrangementofanabsolutelyconvergentserieshasthesamesum,but
thatconditionallyconvergentseriesfail,spectacularly,tohavethisproperty.
Theorem4.3.24
If
P
1
nD1
b
n
isarearrangementofanabsolutelyconvergentseries
P
1
nD1
a
n
;then
P
1
nD1
b
n
alsoconvergesabsolutely;andtothesamesum:
Proof
Let
A
n
Dja
1
jCja
2
jCCja
n
j and
B
n
Djb
1
jCjb
2
jCCjb
n
j:
Foreachn1,thereisanintegerk
n
suchthatb
1
,b
2
,...,b
n
areincludedamonga
1
,a
2
,
..., a
k
n
,so
B
n
A
k
n
. Sincef
A
n
gisbounded,soisf
B
n
g,andtherefore
P
jb
n
j< 1
(Theorem4.3.8).
Nowlet
A
n
Da
1
Ca
2
CCa
n
; B
n
Db
1
Cb
2
CCb
n
;
AD
X1
nD1
a
n
; and BD
X1
nD1
b
n
:
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