Table Of ContentLecture Notes in Mathematics 2185
Bernard Candelpergher
Ramanujan
Summation
of Divergent
Series
Lecture Notes in Mathematics 2185
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Zurich
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Bernard Candelpergher
Ramanujan Summation
of Divergent Series
123
BernardCandelpergher
LaboratoireJ.A.Dieudonné.CNRS
UniversitédeNice
Côted’Azur,Nice,France
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-63629-0 ISBN978-3-319-63630-6 (eBook)
DOI10.1007/978-3-319-63630-6
LibraryofCongressControlNumber:2017948388
Mathematics Subject Classification (2010): 40D05, 40G05, 40G10, 40G99, 30B40, 30B50, 11M35,
11M06
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Introduction: The Summation of Series
Thestrangesums
CX1
nD0C1C2C3C4C5C6C7C8C9C:::
n(cid:2)0
and
CX1
n3 D0C13C23C33C43C53C63C73:::
n(cid:2)0
appearinphysicsaboutthestudyoftheCasimireffectwhichistheexistenceofan
attractiveforcebetweentwoparallelconductingplatesinthevacuum.
Theseseriesareexamplesofdivergentseriesincontrasttoconvergentseries;the
notionofconvergenceforaserieswasintroducedbyCauchyinhisCoursd’Analyse
inordertoPavoidfrequentmistakesinworkingwithseries.Givenaseriesofcomplex
numbers a ,Cauchyconsidersthesequenceofthepartialsums
n(cid:2)0 n
s0 D 0
s1 D a0
s2 D a0Ca1
:::
sn D a0C:::Can(cid:3)1
v
vi Introduction:TheSummationofSeries
P
andsaysthattheseries a isconvergentifandonlyifthesequence.s /hasa
n(cid:2)0 n n
finitelimitwhenngoestoinfinity.Inthiscasethesumoftheseriesisdefinedby
CX1
a D lim s
n n
n!C1
nD0
P
1
TheclassicalRiemannseries isconvergentforeverycomplexnumbers
n(cid:2)1 ns
suchthatPRe.s/>1anddefinestheRiemannzetafunctiondefinedforRe.s/>1by
(cid:2) Ws7! C1 1.
nD1 ns
Non-convergentseries are divergent series. For Re.s/ (cid:2) 1 the Riemann series
is a divergent series and does not give a finite value for the sums that appear in
the Casimir effect. A possible strategy to assign a finite value to these sums is to
performananalyticcontinuationofthezetafunctionthishasbeendonebyRiemann
(Edwards2001)whofoundanintegralformulafor(cid:2).s/whichisvalidnotonlyfor
PRe.s/ > 1 but also for s 2 Cnf1g. By this method we can assign to the series
nk withk>(cid:3)1thevalue(cid:2).(cid:3)k/;weget,forexample,
n(cid:2)1
X 1
n0 D 1C1C1C1C1C1C:::7!(cid:2).0/D(cid:3)
2
n(cid:2)1
X 1
n1 D 1C2C3C4C5C6C:::7!(cid:2).(cid:3)1/D(cid:3)
12
n(cid:2)1
X
n2 D 1C22C32C42C52C:::7!(cid:2).(cid:3)2/D0
n(cid:2)1
X 1
n3 D 1C23C33C43C53C:::7!(cid:2).(cid:3)3/D
120
n(cid:2)1
:::
Fork D(cid:3)1wehavethe“harmonicseries”
X1 1 1 1 1 1 1 1 1
D1C C C C C C C C C:::
n 2 3 4 5 6 7 8 9
n(cid:2)1
whichiseasilyprovedtobeadivergentseriessincethepartialsumss verify
n
Z Z Z
1 1 2 1 3 1 nC1 1
s D1C C:::C (cid:4) dxC dxC:::C dxDLog.nC1/
n 2 n 1 x 2 x n x
Butthestrategyofanalyticcontinuationofthezetafunctiondoesnotworkinthis
casesince(cid:2) hasapoleatsD1thatislims!1(cid:2).s/D1.
Introduction:TheSummationofSeries vii
Divergent series appear elsewhere in aPnalysis and are difficult to handle; for
example,byusingtheprecedingvaluesof nk,itseemsthat
n(cid:2)1
1 1
(cid:3) (cid:3) D1C2C3C4C5C6C:::C.1C1C1C1C1C1C:::/
12 2
D2C3C4C5C6C7C:::
D.1C2C3C4C5C6C7C:::/(cid:3)1
1
D(cid:3) (cid:3)1
12
Thisabsurdityshowsthatwithdivergentserieswecannotusetheclassicalrulesof
calculation,andforagivenclassofseries,weneedtodefinepreciselysomemethod
ofsummationanditsrulesofcalculation.
Before and after Cauchy, some methods of summation of series have been
introduced by several mathematicians such as Cesaro, Euler, Abel, BoPrel, and
others.Thesemethodsofsummationassigntoaseriesofcomplexnumbers a
n(cid:2)0 n
a number obtained by taking the limit of some mePans of the partial sums sn. For
example,theCesarosummationassignstoaseries n a thenumber
n(cid:2)0 n
XC s1C:::Csn
an Dlimn!C1 (whenthislimitisfinite)
n
n(cid:2)0
FortheAbelsummation,wetake
XA CX1
an D lim.1(cid:3)t/ snC1tn (whenthislimitisfinite)
t!1(cid:3)
n(cid:2)0 nD0
P
wheretheseries n(cid:2)0snC1tnissupposedtobeconvergentforeveryt2Œ0;1(cid:3).Note
thatthisexpressioncanbesimplifiedsince
CX1 CX1 CX1
.1(cid:3)t/ snC1tn Ds1C .snC1(cid:3)sn/tn D antn;
nD0 nD1 nD0
andwehave
XA CX1
a D lim a tn (whenthislimitisfinite)I
n n
t!1(cid:3)
n(cid:2)0 nD0
thisgives,forexample,
XA CX1 1 1
.(cid:3)1/n D lim .(cid:3)1/ntn D lim D
t!1(cid:3) t!1(cid:3)1Ct 2
n(cid:2)0 nD0
viii Introduction:TheSummationofSeries
Theclassicalmethodsofsummationusethesetypesofmeansofpartialsumand
canbebrieflypresentedinthefollowingform.
Let T be a topological space of parameters and l some “limit point” of the
compactification of T (if T D N, then l D C1; if T D Œ0;1(cid:3), then l D 1). Let
.pn.t//n2N be aPfamily of complex sequences indexed by t 2 T such that for all
t2T theseries p .t/isconvergent;thenweset
n(cid:2)0 n
P
XT C1p .t/s
a Dlim PnD0 n n
n t!l C1p .t/
n(cid:2)0 nD0 n
P
when this limit is finite, and in this case, we say that the series a is T-
n(cid:2)0 n
summable.
AtheoremofToeplitz(Hardy1949)givesnecessaryandsufficientconditionson
thefamily.pn.t//n2Ntoensurethatincaseofconvergencethissummationcoincides
withtheusualCauchysummation.
Thesesummationmethodsverifythelinearityconditions
XT XT XT
.a Cb /D a C b
n n n n
n(cid:2)0 n(cid:2)0 n(cid:2)0
XT XT
Ca DC a foreveryconstantC2C
n n
n(cid:2)0 n(cid:2)0
andtheusualtranslationproperty
XT XT
an Da0C anC1
n(cid:2)0 n(cid:2)0
ThislaPstpropertywhichseemsnaturalisin factveryrestrictive.Forexample,the
series 1can’tbeT-summablesincethetranslationpropertygivesanabsurd
n(cid:2)1
relation
XT XT
1D1C 1
n(cid:2)0 n(cid:2)0
P
Thus, if we need a method of summation such that the sums T nk are
n(cid:2)0
well defined for any integer k, then we must abandon the translation property
requirement and find a way to define summation procedures other than the way
ofthe“limitofmeansofpartialsums.”
Thiscanbedonebyusingasortofgeneratingfunctionforthetermsoftheseries.
Itisbasedonthefollowingalgebraicframework(Candelpergher1995).LetEbea
C-vectorspace(ingeneralaspaceoffunctions)equippedwithalinearoperatorD
Introduction:TheSummationofSeries ix
andalinearmapv0 W E ! C.Givenasequenceofcomplexnumbers.an/n(cid:2)0,we
callanelementf 2Eageneratorofthissequenceif
an Dv0.Dnf/
Wecanwriteformally
X X X
an D v0.Dnf/Dv0. Dnf/Dv0..I(cid:3)D/(cid:3)1f/I
n(cid:2)0 n(cid:2)0 n(cid:2)0
thusifRverifiestheequation
.I(cid:3)D/RDf; (1)
thenweget
X
an Dv0.R/
n(cid:2)0
Of course, such an algebraic definition of summation needs some hypotheses,
especially to assure uniqueness of the solution of Equation (1); this is presented
inChap.5.
It is easy to see that the Cauchy summation is a special case of this algebraic
formalism: we take E as the vector space of convergent complex sequences
uD.un/n(cid:2)0and
D W .un/7!.unC1/
v0 W .un/7!u0
Inthiscasethegeneratorofasequenceofcomplexnumbers.an/n(cid:2)0ispreciselythis
sequencef D.an/n(cid:2)0since.Dnf/k DakCn:IfwesetRD.rn/n(cid:2)0,Eq.(1)becomes
thedifferenceequation
rn(cid:3)rnC1 Dan
The solution of this equation is defined up to an arbitrary constant; thus to get a
uniquesolution,weneedtoimposeaconditionon.rn/n(cid:2)0.Since
r0(cid:3)rn Da0C:::Can(cid:3)1;
weseethatifweaddthecondition
lim r D0;
n
n!C1
