A MASTER THEOREM OF SERIES AND AN EVALUATION OF A CUBIC HARMONIC SERIES. 1. Introduction

Transcription

1 Joural of Classical Aalysis Volume 0, umber 07, doi:0.753/jca-0-0 A MASTER THEOREM OF SERIES AD A EVALUATIO OF A CUBIC HARMOIC SERIES COREL IOA VĂLEA Abstract. I the actual paper we preset ad prove a ew powerful theorem, a Master Theorem of Series, with importat implicatios ad umerous applicatios i the area of calculatio of series with the geeralized harmoic umbers. Usig the metioed theorem, we calculate oe of the classical advaced cubic harmoic series by elemetary series maipulatios.. Itroductio The cetral theorem of the paper, a Master Theorem of Series, saysthatif is a positive iteger with M m m m, adm are real umbers, where lim m0, the the followig double equality holds M m H j m j, j m j where H... deotes the th harmoic umber. Usig the stated Master Theorem of Series we will prove elemetarily that 3 H 93 8 ζ6 5ζ 3, a classical result i the area of calculatio of Euler sums which i [5] ca be foud evaluated by meas of complex aalysis. Also similar series were previously calculated by meas of real methods see [] ad [], usig a couple of special logarithmic itegrals, elemetary maipulatios of series ad the well-ow Euler s idetity i 6 see[6]. With the ew approach, The Master Theorem of Series will allow us to get the desired results without usig itegrals, but oly by usig elemetary maipulatios of series ad the well-ow Euler s idetity i 6. We metio that i the evaluatio of the cubic Euler sum we also employ auxiliary results which are ot ew ad they exist i the mathematical literature. The series preset i Lemma 5 ad Lemma 6 are ow ad evaluated i [5], ad the Corollary ca be foud evaluated i [3]. We state below the first theorem we are goig to prove. Mathematics subject classificatio 00: 40G0, 40A05. Keywords ad phrases: Abel s summatio formula, Euler sums, harmoic umbers, master theorem of series, Riema zeta fuctio. c D l,zagreb Paper JCA

2 98 C. I. VĂLEA THEOREM. A Master Theorem of Series If is a positive iteger with M m m m, ad m are real umbers, where lim m0, the the followig double equality holds we get M m H j m j. j Proof. Cosiderig the partial sum of the series ad usig that j M j j, j j j j j j For the first ier sum i, we have M j m j M j j M j M j M M m j j M M m j j j M M m j j j M M j j M j The, by pluggig i, we obtai that M m j j m H M M j j m M j j. M j m j M j j m j j. 3 m j.

3 A MASTER THEOREM OF SERIES AD A EVALUATIO OF A CUBIC HARMOIC SERIES 99 Lettig ow i 3, ad usig the Stolz Cesàro theorem to show that 0, we obtai that M m H m j lim M j H m H m j m j. j j j m j m j m j COROLLARY. Let be a positive iteger. The followig equality holds H H p H, p p H i i i p p i i ζih pi, p, where H p p p is the th harmoic umber of order p. Proof. The result is obtaied immediately if usig our Master Theorem of Series, the secod equality, where we set M H p, m p. COROLLARY. Let be a positive iteger. The followig equality holds H H3 3ζH 3H H 3 H 3 where H p p p is the th harmoic umber of order p. i Proof. By employig The Master Theorem of Series, thefirst equality, where we set M H ad mh H,weget H H H0 H H H j H H j j j j, j H / j H j H i i,

4 00 C. I. VĂLEA ad maig use of Corollary, the case p, we get H H j j H j H j H j H j j j H3 3H H 3 j j H H 3 j H 3 H3 3H H 3 ζ j j H j j H3 3H H 3 ζ H H 3 H j j j H3 3ζH 3H H H 3 H j 3 j j, where i the calculatios we made use of Lemma, ad our proof is fialized. LEMMA. Let, p be a positive iteger. The followig equality holds H p p Hp H p, where H p p p is the th harmoic umber of order p. Proof. The result is straightforward by Abel s summatio formula see [4, p. 55] where we set a / p ad b H p. LEMMA. Let be a positive iteger. The followig equality holds H H 3 H3 3H H H 3, where H p p p is the th harmoic umber of order p.

5 A MASTER THEOREM OF SERIES AD A EVALUATIO OF A CUBIC HARMOIC SERIES 0 Proof. Usig Abel s summatio formula see [4, p. 55] with a ad b H H,wehave H H H H H H H H H H H / H H H H H H H H H H H / H H H H H H H. 4 The, we apply Abel s summatio formula for b H, ad the we get H,whereweseta / ad H H H H H H H H H H H H H H 3 H H 3 H H The, by combiig the results from 4ad5, we obtai H H H H H H H H H H H3 3 H H H H H 3 3H H H 3,

6 0 C. I. VĂLEA whece we get ad our lemma is proved. H H 3 H3 3H H H 3, LEMMA 3. Let be a positive iteger. The followig equality holds H H H3 3H H 3 H 3 where H p p p is the th harmoic umber of order p. Proof. The result is obtaied immediately by combiig Corollary ad Corollary. LEMMA 4. The followig equality holds H 7 4 ζ4. Proof. Usig the case p of Corollary ad multiplyig both sides of the equality by /,weget H H H. Summig both sides of the equality above from to, we obtai that H whece we get that H H 5 ζ4 H H H H H H /H H / H, H H 3 H 5 4 ζ4 H 5 4 ζ4, 5 ζ4 ζ ζ47 4 ζ4,

7 A MASTER THEOREM OF SERIES AD A EVALUATIO OF A CUBIC HARMOIC SERIES 03 where above we made use of the well-ow liear Euler sum idetity H ζ ζ ζ,,, 6 ad Lemma, the case p, ad the proof of the lemma is complete. LEMMA 5. The followig equality holds H 4 ζ 3 ζ6 3. Proof. To calculate our series, we first start with a slightly differet series ζ Cosiderig the partial fractio decompositio i the right-had side of , we get 4 that leads immediately to ζ6 ζ 3, 8 where we used that due to symmetry 4,the 4 the fact that ζ 3 ζ6 which is straightforward if otig that ζ 3,adfially the idetity i 6. 3 Therefore, based upo the result i 8, we have 3 5 ζ6 ζ ζ H ζ 4 H ζ6 H 4, whece we obtai that H 4 ad the proof of the lemma is complete. ζ 3 ζ6 3,

8 04 C. I. VĂLEA LEMMA 6. The followig equality holds H ζ6 ζ 3, where H deotes the th harmoic umber. Proof. Usig the case p of Corollary ad the multiplyig both sides of the equality by / 3,wehave H 3 H H 4. The, taig the sum over both sides of the relatio above from to, we get that H 4 H whece we obtai that 4 H ζ3 ζ3 ζ3 ζ3 H 3 3 H ζ H 3 H H 4 H / H / ζ 3 H / H 4 H H / ζ H / 3 H ζ3 3 ζ 3 ζ H 5 H 4 H /H 4 3 ζ ζ6 H 4 3 ζ ζ6 H 4, 4 H ζ6 3ζ 3 H ζ6 ζ 3, where i the calculatios we made use of Euler s idetity i 6 ad Lemma 5. ow we state ad prove the secod theorem.

9 A MASTER THEOREM OF SERIES AD A EVALUATIO OF A CUBIC HARMOIC SERIES 05 THEOREM. A advaced cubic harmoic series The followig equality holds H ζ6 5ζ 3, where H deotes the th harmoic umber. Proof. We mae use of Lemma 3 H H H3 3H H 3 H 3 which we multiply by /, ad summig both sides from to, we get that H H 3 or if chagig the summatios order, we obtai Sice we have that the we get 3 ζ ζ H H 3 3 H H H H H H 3 H 3 H H H. 3 3 H H ζ H, H ζ H H ζ H H 3 H / ζ H 3 3 H H 3 H 3 3, H 3 3. H / H 3

10 06 C. I. VĂLEA H / H 3 H H ζ ζ 3 ζ 4 H ζ ζ 3 4 H H 5 H H 3 H 5 H H ζ ζ 3 ζ 4 ζ 3 H H 4 H 5 H H H 3 5 ζ ζ6ζ H H ζ H 4 97 ζ6 3ζ 3 97 ζ6 3ζ 3 whece we obtai that H H H 3 97 H 3 3 H ζ6 3ζ 3 3 H H 3 H H 3, H 3 H H 4 ζ 4 H ζ6 5ζ 3, where i the calculatios we made use of Euler s idetity i 6, the cases p,3 of Lemma, Lemma 4 ad Lemma 6. Acowledgemet. The author thas the referee for carefully readig the paper ad maig valuable commets throughout the cotet that led to the preset versio of the paper. Also the author thas the JCA joural team ivolved i the phases the paper passed through. REFERECES [] C. I. VĂLEA AD O. FURDUI, Revivig the quadratic series of Au Yeug, JCA6, 05, o., 3 8.

11 A MASTER THEOREM OF SERIES AD A EVALUATIO OF A CUBIC HARMOIC SERIES 07 [] C. I. VĂLEA, A ew proof for a classical quadratic harmoic series, JCA8, 06, o., 55 6 [3] A. SOFO, Harmoic umber sums i higher powers, Joural of Mathematical Aalysis, Volume Issue 0, 5. [4] D. D. BOAR AD M. J. KOURY, Real Ifiite Series, MAA, Washigto DC, 006. [5] P. FLAJOLET AD B. SALVY, Euler sums ad cotour itegral represetatios, Experimet. Math, 7 998, [6] H. M. SRIVASTAVA, J. CHOI, Zeta ad q-zeta Fuctios Ad Associated Series Ad Itegrals, Elsevier, Amsterdam 0. Received Jue 0, 06 Corel Ioa Vălea Teremia Mare, r. 63 Timis, , Romaia corel00 ro@yahoo.com Joural of Classical Aalysis jca@ele-math.com