ON THE SPEED OF CONVERGENCE OF THE SEQUENCES

Transcription

1 Joural of Sciece ad Arts Year, No (, pp 5-60, 0 ORIGINAL PAPER ON THE SPEED OF CONVERGENCE OF THE SEQUENCES ANDREI VERNESCU Mauscript received:00; Accepted paper: 00 Published olie: 000 Abstract The study of ay otrivial covergece of a sequece of real umbers coducts to the problem of fidig the limit but also to the problem of the speed of this covergece This speed of covergece is characterized by the first iterated limit (that supposes the existece of a fuctio of atural variable that teds to zero for tedig to ifiity, a fuctio that appears as the first term of a sequeces of such fuctios A sigificat characterizatio of the first iterated limit may be give by a two sided estimate, from which the first iterated limit results immediately Keywords: Sequeces, series, the costat of Napier, the harmoic sum, the costat of Euler, two sided estimate, iterated limits, asymptotic equivalece Mathematics Subject Classificatio 00: 6D5; 0B0; F05; 40A05; 40A60 INTRODUCTION Whe we study ay otrivial covergece of a sequece of real umbers (a, let be this a ( a, we are iterested ot oly by the obtaiig of the limit, but also by obtaiig of some iformatio about the speed of covergece of the sequece to its limit This speed ca be characterized by the first iterated limit, which is cosidered respectig a fuctio of atural variable u ( [i e of a sequece u ( ], such that ( lim a u ( l ( a where l is a real umber (fiite (The fuctio u ( belogs to a sequece of fuctios A characterizatio of this type also ca be obtaied from a two sided estimate of the form u ( ( l ( a u ( ( l (, ( a with 0 ( (, where ( 0, for (which automatically gives that also we have ( 0, for A two sided estimate of the form ( is more rich i iformatio tha a relatio of ( type, because ( permits us to deduce the equality (, but the coverse affirmatio does t hold A more cocise writig, which shows the existece of a relatio of ( type, but which also cotais less iformatio tha ( is the oe that uses the symbol O, of Ladau, amely Valahia Uiversity of Targoviste, Faculty of Sciece ad Arts, 004 Targoviste, Romaia averescu@gmailcom ISSN: Mathematics Sectio

2 54 O the speed of covergece Adrei Verescu a a O( u ( ( I this paper we will expose some of the two sided estimates of the ( type, obtaied i relatio with several remarkable cocrete sequeces of real umbers THE SEQUENCE WHICH DEFINES THE CONSTANT OF NAPIER AND THE RELATED TO THIS Cocerig the sequece which defies the umber e, also called the costat of Napier or the umber of Euler ad which has the geeral term e ( /, its speed of covergece is described by a two sided estimate metioed i the famous book of G Pólya ad G Szegö [], page 8, amely e e e (4 I G Pólya ad G Szegö [], this result is icluded i a series of other results, also iterestig, but the proof eeds several prelimiary results, which ca be obtaied by certai Taylor-Maclauri expasios I the work [8], published i 988, but ready sice 985, I gave a shorter ad simpler proof for the iequality (4, based oly o the well kow iequality of Hermite-Hadamard (also called of Jese-Hadamard for the covex fuctios a b b a b a ( a ( b ( x dx, applied to the fuctio t ( t / t o the iterval [ x, x ], where x 0, which is l( x l x x x x( x But we also ca cosider other stadard-sequeces related to the umber e, obtaied by modifyig i a atural maer the expoet, replacig it by + p, where p is ay real p (fixed umber, i e cosiderig the sequece of geeral term ( / I [] we also ca fid a classical result of I Schur, amely that if p /, the the sequece is strictly icreasig, ad if p /, the the sequece is strictly decreasig begiig from a certai rak 0 ( p, which depeds o p (i the case p = /, the mootoy begiig accurately sice = Related to the sequece of geeral term f ( /, we have established i [6] the two sided estimate e e e (5 wwwjosaro Mathematics Sectio

3 O the speed of covergece Adrei Verescu 55 Other lifelike ad i a certai sese caoical sequeces, but which coverge to /e are the oe of geeral term ( /, respectively ( / Cocerig these sequeces, I established i my joit papers with C P Niculescu [8] ad [9], the two sided estimatios e (6 e ( e ( e e ( e (7 p Also, cocerig all the covergeces ( / e, we recetly established that the oly which has the speed of / is the oe correspodig to the value p /, ad this is described by the two sided estimatio e / e e ( (8 THE SEQUENCE WHICH DEFINES THE CONSTANT OF EULER Usig the usual otatio for the harmoic sum of order, amely H / / /, the sequece which defies the costat of Euler (also called the costat of Euler-Mascheroi is H l The sequece ( is strictly decreasig ad lower bouded, the it is coverget to a limit which is, by defiitio, the costat of Euler (also deoted sometime by C The speed of covergece is described by the two sided estimate of [7] of 98, (9 The iequality was foud agai ad published i Mathematical Gazette i 99 by R M Youg [6], but the author gave a differet proof A adjacet sequece of ( is the sequece of geeral term H l( ; the term of adjacet sequece of a give mootoic oe is aother sequece which has the same limit as the first, the opposite sese of mootoy ad is geerated by the same aalytic expressio f (, t, but for aother value of the parameter t I our case, f (, t H l( t; The sequece ( correspods to the value t 0 ad the sequece ( correspods to the value t Cocerig the speed of covergece of the sequece (, we established a two sided estimate similar to (9 ISSN: Mathematics Sectio

4 56 O the speed of covergece Adrei Verescu (0 Let s retur ow at the sequece which defies the costat of Euler To improve the speed of covergece to the limit, i e to the costat of Euler, several ew sequeces were defied So, deotig, with D W De Temple, R H l( /, we have the followig two sided estimatio ([] R ( 4( 4 I [] we modified the geeral term of the sequece ( by operatig ot o the argumet of the logarithm, but o the last term of the harmoic sum H, amely /, which was replaced by / So, deotig x l, The speed of covergece is also of the order of /, amely ( x ( Modifyig deeply the argumet of the logarithm, amely defiig the followig iequality holds 48( T H l, 4 T, ( 48 the, the speed of covergece is of the order of / But, for every k N, we ca obtai easy a covergece with the speed of the order of k / by usig the sequece of geeral term k l( k (4 I recet papers, Cristiel Mortici defied several sequeces which coverge to the costat of Euler-Mascheroi ad improved the speed of covergece to it We also metio the works of Alia Sâtămăria regardig this matter wwwjosaro Mathematics Sectio

5 O the speed of covergece Adrei Verescu 57 4 THE SEQUENCE W This sequece is related to the oe which appears i the formula of Wallis Deote by the geeral term of the formula of Wallis, amely ad W 55 7 ( ( (5 5 ( 4 6 (6 I itroduced this otatio (6 i 99, ad I used it i my papers [0], [5], [6] ad others The followig relatio holds W (7 Cocerig, we metio the iequality from D S Mitriović ad P M Vasić [4] the iequality of Wallis, amely ( / (8 ad the iequality of D K Kazarioff ( / ( / 4 (9 I a paper [8] of 985, the regretted professor Laureţiu Paaitopol gave a refiemet of (8, amely (0 ( / 4 / ( / 4 Usig this iequality I established i [5] the speed of covergece of the sequece of the formula of Wallis, amely W 4( 8 4( ( I the paper [0] i cooperatio with Lászlo Tóth, also of 99, we established the asymptotic expasio of the sequece of Wallis W ( ISSN: Mathematics Sectio

6 58 O the speed of covergece Adrei Verescu I this paper [0] we also established the asymptotic expasio of the sequece, amely 5 = ( I the paper [6] I established several geeralizatios of (0 cotaiig more terms uder the square root ad asymptotic expasios which cotiue the formula (0 with more terms 5 THE SEQUENCE OF TRAIAN LALESCU Cosider the sequece of Traia Lalescu, of geeral term ( L (!! It is coverget to the real umber /e; there is a extesive literature related to this, which does t costitute oe of the aims of this paper I would oly metio that the problem of fidig elemetary solutios was raised by Tiberiu Popoviciu i 97 The first two elemetary solutios were give by the regretted professor Alexadru Lupaş i 976 ad by Marcel Ţea i 978 The speed of covergece of this sequece to its limit is give by the followig iequality of [] l L e, (4 e e where is ay real umber greater tha ¼ We also metio a asymptotic expasio of, give i [] L 6 THE SPEED OF CONVERGENCE OF THE GENERALIZED HARMONIC SERIES Let s be a real umber, s Deote s s s ( s The limit of this sequece whe teds to ifiity, which also ca be writte as the sum of a series, defies the celebrated fuctio of Riema, which is very importat i all mathematics The speed of covergece of the sequece to its limit is described by the followig two sided estimate wwwjosaro Mathematics Sectio

7 O the speed of covergece Adrei Verescu 59 ( s ( s ( s ( s, (5 s ( s which ca be foud i the well kow treatise of G M Fihteholtz I this book, the iequality is obtaied by cosiderig the remaider of a certai improper itegral I a paper of 997, I obtaied aother proof for this iequality, based oly o the mootoicity of two adequate sequeces ad o the mea theorem of Lagrage The fuctio of Riema is exteded to the complex variable, s it, ad the most iterestig problems appear i this domai The crucial problem is the celebrated hypothesis of Riema, cocerig the distributio of the otrivial zeros of the fuctio o the critical lie Re(s =/ 7 THE COMPENSED HARMONIC SEQUENCES OF EULER TYPE If the real umber s is ot smaller tha, the the precedet series is diverget A importat discovery of Euler was the fact that by subtractio from the partial sum (s, s (where s (0, of the term ispired exactly by the primitive fuctio of x f ( x / x, we obtai a coverget sequece A remarkable fact is the oe that this covergece remais valid for the adheret case of s ; of course, i this case, the primitive fuctio chages its ature becomig the logarithm ad givig the celebrated costat of Euler, also amed of Euler-Masheroi Deote that the case s /, surely kow by Euler, was revisited by Adrei G Ioachimescu i the first issue of Gazeta Matematică, of 895 Deotig a ad a lim a, I established i 99 the two sided estimate a a The correspodig two sided estimate for the adjacet sequece, of geeral term b was established i [8] 8 CONCLUDING REMARKS All the previous examples show several speeds of covergece of the sequeces to their limits, ivolvig a collectio of these speeds The two sided estimates coduct to the fidig of the first iterated limit accordigly with the asymptotic expasios (respectig a ISSN: Mathematics Sectio

8 60 O the speed of covergece Adrei Verescu certai sequece of fuctios of atural variable where such expasios exist I some cases these limits suggested the first steps i the asymptotic expasios REFERENCES [] De Temple, DW, Amer Math Mothly, 00, 468, 99 [] Kazarioff, DK, Ediburgh Math Notes, 40, 9, 956 [] Lupaş, A,Verescu, A, G M Seria A, 9(4,, 00 [4] Mitriović, DS, Vasić, PM, Aalytic Iequalities, Spriger Verlag, Berli Heidelberg New York, 970 [5] Negoi, T, G M Seria A, 5(,, 997 [6] Niculescu, CP, Verescu, A, JIPAM, 5(, art 55, 004 [7] Niculescu, CP, Verescu, A, Gazeta Matematică, 09(4, 45, 004 [8] Paaitopol, L, Gazeta Matematică, 90(9, 9, 985 [9] Pólya, G, Szegö, G, Theorems ad Problems i Aalysis, Spriger Verlag, Berli Heidelberg New York, 970 [0] Tóth, L, Verescu, A, G M Seria A,, 6, 990 [] Tóth, L, Becze, M, Octogo, 4(,, 996 [] Verescu, A, Gazeta Matematică, -, 6, 98 [] Verescu, A, Gazeta Matematică, 0-, 80, 98 [4] Verescu, A, Gazeta Matematică, 5-6, 06, 988 [5] Verescu, A, G M Seria A, (, 7, 99 [6] Verescu, A, Sur l approximatio et le développemet asymptotique de la suite de (!! terme gééral, Lucrările Coferiţei Auale a Societăţii de Ştiiţe (!! Matematice di Româia, Bucuresti, 9 mai- iuie 997, vol, Editori M Becheau, I D Io, AVerescu, Bucureşti, 998, 05- [7] Verescu, A, G M Seria A, 7(4, 7, 999 [8] Verescu, A, O the Similar Order of Covergece of two Adjacet Sequeces Related to (/, WSEAS Tras o Math, 5(, 97, 006 wwwjosaro Mathematics Sectio