SOME OPEN PROBLEMS CONCERNING THE CONVERGENCE OF POSITIVE SERIES 2 1 CONSTANTINP.NICULESCUANDGABRIELT.PRAˇJITURAˇ 0 2 Abstract. WediscusssomeoldresultsduetoAbelandOlivierconcerningthe n convergenceofpositiveseriesandproveasetofnecessaryconditionsinvolving a convergence indensity. J 4 2 1. Introduction ] A Understanding the nature of a series is usually a difficult task. The following C two striking examples can be found in Hardy’s book, Orders of infinity: the series . h 1 at nlnn(lnlnn)2 n 3 m X≥ convergesto38.43...,butdoesitsoslowthatoneneedstosumupitsfirst103.14 1086 [ × terms to get the first 2 exact decimals of the sum. In the same time, the series 1 1 v 6 nlnn(lnlnn) n 3 5 X≥ 1 isdivergentbutitspartialsumsexceed10onlyafter1010100 terms. See[17],pp. 60- 5 61. Onpage48ofthe same book,Hardymentionsaninterestingresult(attributed . 1 to De Morgan and Bertrand) about the convergence of the series of the form 0 1 1 2 (MB ) and , k ns n(lnn)(lnlnn) (lnln lnn)s 1 v: nX≥1 nX≥nk ··· kt·im··es i where k is an arbitrarily fixed natural number, s i|s a r{ezal n}umber and nk is a X number largeenoughto ensure that lnln lnn is positive. Precisely,sucha series r ··· a ktimes is convergent if s > 1 and divergent otherwise. This is an easy consequence of | {z } Cauchy’s condensation test (see Knopp [21], p. 122). Another short argument is provided by Hardy [18] in his Course of Pure Mathematics, on p. 376. The above discussion makes natural the following problem. Problem 1. What decides if a positive series is convergent or divergent? Is there any universal convergence test? Is there any pattern in convergence? This is an old problem who received a great deal of attention over the years. Importantprogresswasmadeduringthe19thCenturybypeoplelikeA.-L.Cauchy, N. H. Abel, C. F. Gauss, A. Pringsheim and Du Bois-Reymond. In the last fifty Date:Jan19,2012. 2000 Mathematics Subject Classification. Primary37A45,40A30;Secondary 40E05. Key words and phrases. positiveseries,setofzerodensity,convergence indensity. 1 2 CONSTANTINP.NICULESCUANDGABRIELT.PRAˇJITURAˇ years the interest shifted toward combinatorial aspects of convergence/divergence, althoughpapers containing new tests of convergencecontinue to be published. See forexample[2]and[23]. Thispaper’spurposeistodiscusstherelationshipbetween theconvergenceofapositiveseriesandtheconvergencepropertiesofthesummand sequence. 2. Some history WestartbyrecallinganepisodefromthebeginningofAnalysis,thatmarkedthe moment when the series of type (MB ) entered the attention of mathematicians. k M. Goar [14] has written the story in more detail. In 1827, L. Olivier [26] published a paper claiming that the harmonic series representsakindof“boundary”casewithwhichotherpotentiallyconvergentseries ofpositive terms could be compared. Specifically, he assertedthat a positive series a whose terms are monotone decreasing is convergent if and only if na 0. n n → One year later, Abel [1] disproved this convergence test by considering the case oPf the (divergent) positive series 1 . In the same Note, Abel noticed two n 2 nlnn other important facts concerning the≥convergence of positive series: P Lemma 1. There is no positive function ϕ such that a positive series a whose n terms are monotone decreasing is convergent if and only if ϕ(n)a 0. In other n →P words, there is no “boundary” positive series. Lemma 2. If a is a divergent positive series, then the series an is n Pnk=1ak also divergent.PAs a consequence, for each divergent positive serieXs th(cid:16)ere is alw(cid:17)ays another one which diverges slower. A fact which was probably known to Abel (although it is not made explicit in his Note) is that the whole scale of divergent series 1 (A) for k=1,2,3,... n(lnn)(lnlnn) (lnln lnn) nX≥nk ··· ··· ktimes comes from the harmonic series 1,|by s{uzcces}sive application of Lemma 2 and n the following result on the generalized Euler’s constant. P Lemma 3. (C. Maclaurin and A.-L. Cauchy). If f is positive and strictly decreasing on [0, ), there is a constant γ (0,f(1)] and a sequence (E (n)) with f f n ∞ ∈ 00 such that P n ε a z < k k (cid:12) (cid:12) 4 (cid:12)k=Xm+1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) foreveryn>m.We willestimatea (z + +z )byusingAbel’s identity. In n m+1 n ··· fact, letting S =a z + +a z , n m+1 m+1 n n ··· 4 CONSTANTINP.NICULESCUANDGABRIELT.PRAˇJITURAˇ we get 1 1 a (z + +z ) =a a z + + a z n m+1 n n m+1 m+1 n n | ··· | a ··· a (cid:12) m+1 n (cid:12) 1 (cid:12)1 1 (cid:12) (cid:12) (cid:12) =an Sm+1+ (cid:12) (Sm+2 Sm+1)+ + (S(cid:12)n Sn 1) (cid:12)am+1 am+2 − ··· an − − (cid:12) (cid:12) 1 1 1 1 1 (cid:12) (cid:12) (cid:12) =an (cid:12) Sm+1+ + Sn 1+ Sn(cid:12) (cid:12)(cid:18)am+1 − am+2(cid:19) ··· (cid:18)an−1 − an(cid:19) − an (cid:12) (cid:12) εa 1 1 1 1 1 (cid:12) (cid:12) n (cid:12) (cid:12) + + + (cid:12) ≤ 4 a − a ··· a − a a (cid:18)(cid:18) m+2 m+1(cid:19) (cid:18) n n−1(cid:19) n(cid:19) εa 2 1 ε n = < . 4 a − a 2 (cid:18) n n+1(cid:19) Since lim a =0, one may choose an index N(ε)>m such that n n →∞ ε a (z + +z ) < n 1 m | ··· | 2 for every n>N(ε) and thus a (z + +z ) a (z + +z ) + a (z + +z ) <ε n 1 n n 1 m n m+1 n | ··· |≤| ··· | | ··· | for every n>N(ε). (cid:3) In2003,T.Sˇala´tandV.Toma[27]madetheimportantremarkthatthemonotoni- city condition in Theorem 1 can be dropped if the convergence of (na ) is weak- n n ened: Theorem 3. If a is a convergent positive series, then na 0 in density. n n → In order to expPlain the terminology,recall that a subset A of N has zero density if A 1,...,n d(A)= lim | ∩{ }| =0, n n →∞ positive lower density if A 1,...,n d(A)=liminf| ∩{ }| >0, n n →∞ and positive upper density if A 1,...,n d¯(A)=liminf| ∩{ }| >0. n n →∞ Here stands for cardinality. |·| Wesaythatasequence(x ) ofrealnumbersconverges indensity toanumberx n n (denotedby(d)-lim x =x)ifforeveryε>0thesetA(ε)= n: x x ε n n n →∞ { | − |≥ } has zero density. Notice that (d) lim x = x if and only if there is a subset n n J of N of zero density such that − →∞ lim a =0. n n n→/∞J ∈ This notion can be traced back to B. O. Koopman and J. von Neumann ([22], pp. 258-259),who proved the integral counterpart of the following result: POSITIVE SERIES 5 Theorem 4. For every sequence of nonnegative numbers, n 1 lim a =0 (d)- lim a =0. k n n n ⇒ n →∞ k=1 →∞ X The converse works under additional hypotheses, for example, for bounded se- quences. Proof. Assuming lim 1 n a =0, we associate to each ε>0 the set A = n N:a ε . Sinnc→e∞ n k=1 k ε n { ∈ ≥ } P n 1,...,n A 1 a ε k |{ }∩ | n ≤ n ε k=1 X n 1 a 0 as n , k ≤ εn → →∞ k=1 X we infer that each of the sets A has zero density. Therefore (d)-lim a =0. ε n n →∞ Suppose now that (a ) is bounded and (d)-lim a = 0. Then for every n n n n →∞ ε>0 there is a set J of zero density outside which a <ε. Since n n 1 1 1 a = a + a k k k n n n kX=1 k∈{1X,...,n}∩J k∈{1X,...,n}\J 1,...,n J |{ }∩ | supa +ε k ≤ n ·k N ∈ and limn→∞ |{1,...n,n}∩J| =0, we conclude that limn→∞ n1 nk=1ak =0. (cid:3) Remark 2. Theorem 4 is related to the Tauberian theoryP, whose aim is to provide converses to the well known fact that for any sequence of complex numbers, n 1 lim z =z lim z =z. n k n ⇒n n →∞ →∞ k=1 X RecallherethefamousHardy-LittlewoodTauberiantheorem: If z z =O(1/n) n n 1 and | − − | n 1 lim z =z, k n n →∞ k=1 X then lim z =z. See [19], Theorem 28. n n →∞ The aforementioned result of Sˇala´t and Toma is actually an easy consequence of Theorem 4. Indeed, if a is a convergent positive series, then its partial n sums S = n a constitutes a convergent sequence with limit S. By Cesàro’s n k=1 k P Theorem, P S + +S 1 n 1 lim ··· − =S, n n →∞ whence a +2a + +na S + +S 1 2 n 1 n 1 lim ··· = lim Sn ··· − =0. n n n − n →∞ →∞(cid:18) (cid:19) According to Theorem 4, this fact is equivalent to the convergence in density of (na ) to 0. n n 6 CONSTANTINP.NICULESCUANDGABRIELT.PRAˇJITURAˇ In turn, the result of Sˇala´t and Toma implies Olivier’s Theorem. Indeed, if the sequence (a ) is decreasing, then n a +2a + +na (1+2+ +n)a (n+1)a 1 2 n n n ··· ··· = n ≥ n 2 which implies that if a +2a + +na 1 2 n lim ··· =0 n n →∞ then lim na =0. n n If a isaconvergentpositiveseries,thensois a ,wheneverϕ:N Nis n ϕ(n) → abijective map. This impliesthatna 0indensity(a conclusionthatdoesn’t ϕ(n) P → P work for usual convergence). The monograph of H. Furstenberg [13] outlines the importance of convergence in density in ergodic theory. In connection to series summation, the concept of convergence in density was rediscovered (under the name of statistical convergence) by Steinhaus [28] and Fast [12] (who mentioned also the first edition of Zygmund’smonograph[31], published in Warsawin 1935). Apparently unawareof the Koopman-von Neumann result, Sˇala´t and Toma referred to these authors for the roots of convergence in density. Atpresentthereisalargeliteratureaboutthisconceptanditsmanyapplications. We only mention here the recent papers by M. Burgin and O. Duman [7] and P. Therań [30]. 3. An extension of Sˇala´t - Toma Theorem In this section we will turn our attention toward a generalization of the result of Sˇala´t and Toma mentioned above. This generalization involves the concepts of convergence in density and convergence in lower density. A sequence (x ) of real n n numbers converges in lower density to a number x (abbreviated, (d)-lim x = n n →∞ x) if for every ε>0 the set A(ε)= n: x x ε has zero lower density. n { | − |≥ } Theorem 5. Assume that a is a convergent positive series and (b ) is a n n n nondecreasing sequence of positive numbers such that ∞ 1 = . Then P n=1 bn ∞ (d)- lim a b =0, P n n n →∞ and this conclusion can be improved to (d)- lim a b =0, n n n →∞ provided that inf n >0. n bn Animmediateconsequenceisthefollowingresultaboutthespeedofconvergence to 0 of the general term of a convergent series of positive numbers. Corollary 1. If a is a convergent series of positive numbers, then for each n k N, ∈ P (D ) (d)- lim n(lnn)(lnlnn) (lnln lnn)a =0. k n  ··· ··· n →∞ k times   The proof of Theorem 5 is based on two tec|hnica{zl lem}mas: POSITIVE SERIES 7 Lemma4. Supposethat(c ) isanonincreasingsequenceofpositivenumberssuch n n that ∞n=1cn = ∞ and S is a set of positive integers with positive lower density. Then the series c is also divergent. P n∈S n Proof. By our hyPpothesis there are positive integers p and N such that S 1,...,n 1 | ∩{ }| > n p whenever n N. Then S 1,...,kp >k for every k N/p, which yields ≥ | ∩{ }| ≥ ∞ ∞ c = c c n nk ≥ kp n S k=1 k=1 X∈ X X 1 ∞ = pc kp p k=1 X 1 [(c + +c )+(c + +c )+ ] p 2p 1 2p 3p 1 ≥ p ··· − ··· − ··· 1 ∞ = c = . k p ∞ k=p X (cid:3) Our second lemma shows that a subseries 1 of the harmonic series is n S n divergent whenever S is a set of positive integers w∈ith positive upper density. P Lemma5. IfS isaninfinitesetofpositiveintegersand(a ) isanonincreasing n n S positive sequence such that a < and inf nc :n ∈S = α > 0, then S n S n ∞ { n ∈ } has zero density. ∈ P Proof. Accordingtoourhypotheses,theelementsofS canbecountedask 0 arbitrarily fixed we denote S = n:a b ε . ε n n { ≥ } Then 1 > a , n ∞ n∈Sε ≥ n∈Sε bn whence by Lemma 4 it followXs that S hasXzero lower density. Therefore (d)- ε lim a b =0. When inf n =α>0, then n→∞ n n n bn 1 1 > α ∞ n∈Sε bn ≥ n∈Sε n X X 8 CONSTANTINP.NICULESCUANDGABRIELT.PRAˇJITURAˇ sobyLemma5weinferthatS haszerodensity. Inthiscase,(d)-lim a b =0. ε n n n (cid:3) →∞ 4. Convergence associated to higher order densities Theconvergenceinlowerdensityisveryweak. Abetterwaytoformulatehigher order Sˇala´t-Toma type criteria is to consider the convergence in harmonic density. We will illustrate this idea by proving a non-monotonic version of Remark 1. The harmonic density d is defined by the formula h n 1 χ (k) A d (A)= lim , h n lnn k →∞ k=1 X andthelimitinharmonicdensity,(d )-lim a =ℓ,meansthateachofthe sets h n n →∞ n: a ℓ ε has zero harmonic density, whenever ε>0. Since n { | − |≥ } d(A)=0 implies d (A)=0, h (see [16], Lemma 1, p. 241),it follows that the existence of limit in density assures the existence of limit in harmonic density. The harmonic density has a nice application to Benford’s law, which states that inlistsofnumbersfrommanyreal-lifesourcesofdatatheleadingdigitisdistributed in a specific, non-uniform way. See [8] for more details. Theorem 6. If a is a convergent positive series, then n P (dh)- lim (nlnn)an =0. n →∞ Proof. WestartbynoticingthefollowinganalogueofLemma5: If(b ) isapositive n n sequence such that (nb ) is decreasing and n n inf(nlnn)b =α>0, n then every subset S of N for which b < has zero harmonic density. To provethis assertion,it suffices ton∈cSonsnider∞the case where S is infinite and to P show that 1 (H) lim nb =0. n x k S 1,...,n k →∞(cid:18) ∈ ∩{ } (cid:19) X ThedetailsareverysimilartothoseusedinLemma5,andthustheyareomitted. Having (H) at hand, the proof of Theorem 6 can be completed by considering for each ε>0 the set S = n:(nlnn)a ε . ε n { ≥ } Since 1 ε a < , n n Sε nlnn ≤ n Sε ∞ ∈ ∈ by the aforementioned anXalogue of LemmXa 5 applied to b = 1/(nlnn) we infer n that S has zero harmonic density. Consequently (d )-lim (nlnn)a =0, and ε h x n the proof is done. →∞ (cid:3) An integral version of the previous theorem can be found in [24] and [25]. One might think that the fulfilment of a sequence of conditions like (D ), for k all k N, (or something similar) using other series, is strong enough to force the ∈ convergenceof a positive series a . That this is not the case was shown by Paul n Du Bois-Raymond[6] (see also[21], Ch. IX, Section 41)who provedthat for every P POSITIVE SERIES 9 sequence of divergent positive series, each divergent essentially slower than the previous one, it is possible to construct a series diverging slower than all of them. Under these circumstances the following problem seems of utmost interest: Problem2. Findanalgorithm todeterminewhetherapositiveseriesisconvergent or not. 5. The relevance of the harmonic series Surprisingly, the study of the nature of positive series is very close to that of subseries of the harmonic series 1. n Lemma 6. If (an)n is a unbounPded sequence of real numbers belonging to [1, ), ∞ then the series 1 and 1 have the same nature. an an ⌊ ⌋ Proof. TheconvPergenceofPtheseries 1 followsfromtheconvergenceoftheseries an 1 , by the comparison test. In fact, a a , whence 1 1 . an P n ≥⌊ n⌋ an ≥ an C⌊on⌋versely,if the series 1 is convergent,thensoarethe⌊ se⌋ries 1 and P an an +1 1 . This is a consequence of the inequalities ⌊ ⌋ an ( an +1) P P ⌊ ⌋ ⌊ ⌋ P 1 1 1 , a ( a +1) ≤ a +1 ≤ a n n n n ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ and the comparison test. Since 1 1 1 = + a a +1 a ( a +1) n n n n ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ we conclude that the series 1 is convergenttoo. (cid:3) an ⌊ ⌋ By combining Lemma 5 aPnd Lemma 6 we infer the following result: Corollary 2. If (a ) is a sequence of positive numbers whose integer parts form n n a set of positive upper density, then the series 1 is divergent. an The converse of Corollary 2 is not true. APcounterexample is provided by the series 1, of inverses of prime numbers, which is divergent (see [3] or [10] p=prime p for a short argument). According to an old result due to Chebyshev, if π(n) = P p n:p prime , then |{ ≤ }| 7 π(n) 9 < < 8 n/lnn 8 and thus the set of prime numbers has zero density. The following estimates of the kth prime number, k(lnk+lnlnk 1) p k(lnk+lnlnk) for k 6, k − ≤ ≤ ≥ which are made available by a recent paper of P. Dusart [9], shows that the speed of divergence of the series 1 is comparable with that of 1 . p=prime p k(lnk+lnlnk) Lemma 6 suggests that the nature of positive series 1 could be related to P an P some combinatorialproperties of the sequence ( a ) (of natural numbers). ⌊ n⌋ n P Problem 3. Given an increasing function ϕ:N (0, ) with lim ϕ(n)= , n we define the upper density of weight ϕ by the for→mula∞ →∞ ∞ A [1,n] d¯ (A)=limsup| ∩ |. ϕ ϕ(n) n →∞ 10 CONSTANTINP.NICULESCUANDGABRIELT.PRAˇJITURAˇ Does every subset A N with d¯ (A) > 0 generate a divergent subseries n/lnn ⊂ 1 of the harmonic series? n A n W∈hat about the case of other weights P n/[(lnn)(lnlnn) (lnln lnn)]? ··· ··· k times This problem seems important in connect|ion {wzith t}he following long-standing conjecture due to P. Erdo¨s: Conjecture 1. (P. Erdo¨s). Ifthesumof reciprocals of aset Aof integers diverges, then that set contains arbitrarily long arithmetic progressions. This conjecture is still open even if one only seeks a single progressionof length three. However, in the special case where the set A has positive upper density, a positive answered was provided by E. Szemerédi [29] in 1975. Recently, Green and T. Tao [15] proved Erdo¨s’ Conjecture in the case where A is the set of prime numbers, or a relatively dense subset thereof. Theorem 7. Assuming the truth of Erdo¨s’ conjecture, any unbounded sequence (a ) ofpositive numberswhosesumofreciprocals 1 is divergentmustcontain n n n an arbitrarily long ε-progressions, for any ε>0. P By an ε-progressionof length n we mean any string c ,...,c such that 1 n c a kr <ε k | − − | for suitable a,r R and all k=1,...,n. ∈ The converse of Theorem 7 is not true. A counterexample is provided by the convergentseries ∞n=1 10n1+1 +···+ 10n1+n . It seems to us that wh(cid:16)at is relevantin the m(cid:17)atter of convergenceis not only the P existenceofsomeprogressionsbutthenumberofthem. Webelievenotonlythatthe divergentsubseriesofthe harmonicserieshaveprogressionsofarbitrarylengthbut thattheyhaveahugenumberofsuchprogressionsandofarbitrarilylargecommon differences. Notice that the counterexample above contains only progressions of commondifference1(orsubprogressionsofthem). HardyandLittlewood’sfamous paper [20] advanced the hypothesis that the number of progressions of length k is asymptotically of the form C n2/lnkn, for some constant C . k k References [1] N.H.Abel,NotesurlememoiredeMr.L.OlivierNo.4dusecondtomedecejournal,ayant pour titre ’Remarques sur les series infinies et leur convergence’, Journal fu˝r die reine und angewandte Mathematik, 3 (1828), 79-81. Available fromthe Göttinger Digitalisierungszen- trumathttp://gdz.sub.uni-goettingen.de/no cache/dms/load/toc/?IDDOC=238618 [2] S.A.Ali,ThemthRatioTest: NewConvergenceTestsforSeries,TheAmericanMathemat- ical Monthly 115(2008), No.6,514-524. [3] T.M.Apostol,WhatIstheMostSurprisingResultinMathematics? PartII,MathHorizons, 4(1997), No.3,26-31. [4] T.M.Apostol,AnElementaryViewofEuler’sSummationFormula,The American Mathe- matical Monthly,106(1999), No.5,409-418. [5] J.MarshallAsh,NeitheraWorstConvergent SeriesnoraBestDivergentSeriesExists,The College Mathematics Journal 28(1997, No.4,296-297. [6] P. Du Bois-Reymond, Eine neue Theorie der Convergenz und Divergenz von Reihen mit positivenGliedern,Journal fu˝rdie reine und angewandte Mathematik 76(1873), 61-91.

Some open problems concerning the convergence of positive series

Similar Some open problems concerning the convergence of positive series