You ad Che Joural of Iequalities ad Applicatios 08) 08:7 https://doi.org/0.86/s3660-08-670-6 R E S E A R C H Ope Access A ew sequece coverget to Euler Mascheroi costat Xu You * ad Di-Rog Che * Correspodece: youxu@bipt.edu.c Departet of Matheatics ad Physics, Beijig Istitute of Petrocheical Techology, Beijig, P.R. Chia Full list of author iforatio is available at the ed of the article Abstract I this paper, we provide a ew sequece covergig to the Euler Mascheroi costat. Fially, we establish soe iequalities for the Euler Mascheroi costat by the ew sequece. MSC: Y60; A; A0 Keywords: Euler Mascheroi costat; Rate of covergece; Taylor s forula; Haroic sequece Itroductio The Euler Mascheroi costat was first itroduced by Leohard Euler 707 783) i 73 as the liit of the sequece γ ):= l..) There are ay faous usolved probles about the ature of this costat see, e.g., the survey papers or books of Bret ad Ziera [], Dece ad Dece [], Havil [3], ad Lagarias []). For exaple, it is a log-stadig ope proble if the Euler Mascheroi costat is a ratioal uber. A good part of its ystery coes fro the fact that the kow algoriths covergig to γ are ot very fast, at least whe they are copared to siilar algoriths for π ad e. The sequece γ )) N coverges very slowly toward γ,like).uptoow,ay authors are preoccupied to iprove its rate of covergece; see, for exaple, [, ] ad refereces therei. We list soe ai results: l + ) = γ + O ) DeTeple [6]), l 3 + 3 + 7 + + 97 0 + 07 80 0 = γ + O 6) Mortici [3]), l + + ) 8 + 3 = γ + O ) 3 760 Che ad Mortici []). The Authors) 08. This article is distributed uder the ters of the Creative Coos Attributio.0 Iteratioal Licese http://creativecoos.org/liceses/by/.0/), which perits urestricted use, distributio, ad reproductio i ay ediu, provided you give appropriate credit to the origial authors) ad the source, provide a lik to the Creative Coos licese, ad idicate if chages were ade.
You adchejoural of Iequalities ad Applicatios 08) 08:7 Page of 8 Recetly, Mortici ad Che [] provided a very iterestig sequece ν)= ad proved that l + + ) 3 8 9 ) 80 + + + 83 3 ) + + + 3 )3 + + + 93, 3 ) + + 3 ) li ν) γ ) = 796,80 3,783,70..) Hece the rate of the covergece of the sequece ν)) N is. Very recetly, by isertig the cotiued fractio ter ito.), Lu [9] itroduceda class of sequeces r k )) N see Theore )adshowedthat 7 +) < γ r )< 3 7, 3.3) 0 +) < r 3) γ < 0 )..) I fact, Lu [9]alsofouda without proof, ad his works otivate our study. I this paper, startig fro the well-kow sequece γ, based o the early works of Mortici, DeTeple, ad Lu, we provide soe ew classes of coverget sequeces for the Euler Mascheroi costat. Theore For the Euler Mascheroi costat, we have the followig coverget sequece: where Let r)=+ + + l l + a ) l + a ), a =, a =, a 3 =, a = 3 760, a = 60, a 6 = 90,30, a 7 = 896,... r k ):= k l l + a ). For k 7, we have li k+ r k ) γ ) = C k,.)
You adchejoural of Iequalities ad Applicatios 08) 08:7 Page 3 of 8 where C =, C =, C 3 = 3 760, C = 60, C = 90,30, C 6 = 896, C 7 = 09,793,8,00,... Furtherore, for r )adr 3 ), we also have the followig iequalities. Theore Let r ) ad r 3 ) be as i Theore. The +) < γ r )< 3, 3.6) 3 760 +) < r 3) γ < 3 760..7) Reark Certaily, there are siilar iequalities for r k ) k 7); we oit the details. Proof of Theore The followig lea gives a ethod for easurig the rate of covergece. This lea wasfirstusedbymortici[, 6] for costructig asyptotic expasios or acceleratig soe covergeces. For a proof ad other details, see, for exaple, [6]. Lea If the sequece x ) N coverges to zero ad there exists the liit li + s x x + )=l [,+ ].) with s >,the there exists the liit li + s x = l s..) We eed to fid the value a R that produces the ost accurate approxiatio of the for r )= l l + a )..3) To easure the accuracy of this approxiatio, we usually say that a approxiatio.3) is better as r ) γ faster coverges to zero. Clearly, r ) r +)=l + ) + + l + a ) l + a )..) + Developig expressio.) ito a power series expasio i /,we obtai ) r ) r +)= a + ) ) 3 + a + a + O..) 3 Fro Lea we see that the rate of covergece of the sequece r ) γ ) N is eve higher asthe value s satisfies.). By Lea we have
You adchejoural of Iequalities ad Applicatios 08) 08:7 Page of 8 i) If a /, the the rate of covergece of the r ) γ ) N is,sice li r ) γ ) = a 0. ii) If a =/,thefro.)wehave r ) r +)= ) + O. 3 Hece the rate of covergece of the r ) γ ) N is 3,sice li 3 r ) γ ) = := C. We also observe that the fastest possible sequece r )) N is obtaied oly for a = /. We repeat our approach to deterie a to a 7 step by step. I fact, we ca easily copute a k, k, by the Matheatica software.ithispaper,weusethematheatica software to aipulate sybolic coputatios. Let The r k ):= k l l + a )..6) r k ) r k +) = l + ) + + k ) a l + +) k l + a )..7) Hece the key step is to expad r k ) r k + ) ito power series i /. Hereweuse soe exaples to explai our ethod. Step : For exaple, give a to a,fida.defie r ):= l l + a ). By usig the Matheatica software the Matheatica Progra is very siilar to that give further i Reark ; however, it has the paraeter a 8 )weobtai r ) r +)= ) ) ) 38 3 a + 6 8,38 +a + O..8) 7 8 Thefastestpossiblesequecer )) N is obtaied oly for a =. At the sae tie, 60 it follows fro.8)that r ) r +)= ) 8,38 + O..9) 7 8
You adchejoural of Iequalities ad Applicatios 08) 08:7 Page of 8 Therateofcovergeceofr 8 ) γ ) N is 7,sice li 7 r ) γ ) = 90,30 := C. We ca use this approach to fid a k k ). Fro the coputatios we ay the cojecture a + = C,. Now, let us check it carefully. Step :Checka 6 = 90,30 to a 7 = 896. Let a,...,a 6,adr 6 ) be defied as i Theore. Applyig the Matheatica software, we obtai r 6 ) r 6 +)= ) 8 + O..0) 8 9 Therateofcovergeceofr 6 ) γ ) N is 8,sice li 8 r 6 ) γ ) = 896 := C 6. Fially, we check that a 7 = 896 : r 7 ) r 7 +)= ) ) ) 96,93 8 7a 7 + 8,76,800 +8a 7 + O..) 9 0 Sice a 7 = 896 ad li 9 r 7 ) γ ) = 09,793,8,00 := C 7, the rate of covergece of the r 7 ) γ ) N is 9. This copletes the proof of Theore. Reark Fro the coputatios we ca guess that a + = C,. It is a very iterestig proble to prove this. However, it sees ipossible by the provided ethod. 3 Proof of Theore Before we prove Theore, let us give a siple iequality, which follows fro the Herite Hadaard iequality ad plays a iportat role i the proof. Lea Let f be a twice cotiuously differetiable fuctio. If f x)>0,the a+ a f x) dx > f a + /). 3.) By P k x) we deote polyoials of degree k i x such that all its ozero coefficiets are positive; it ay be differet at each occurrece. Let us prove Theore.Notigthatr )=0,weeasilyseethat γ r )= r +) r ) ) = f ), 3.) = =
You adchejoural of Iequalities ad Applicatios 08) 08:7 Page 6 of 8 where f )= + l + + l + a ) l + a + ) + l + a ). ) l + Let D = /. By usig the Matheatica software we have ad f x) D x +) ) a +) = 300 + 739x +,3x +,870x 3 + 8,3x +,936x + 37x 6 x + x) + x)3 + x) + x ) + 8x +x ) f x) D x + ) P 6 x)x ) +,08 = x + x) + x)3 + x) + x ) + 8x +x ) <0. Hece, we get the followig iequalities for x : D x +) < f x)<d x +. 3.3) ) >0 Sice f ) = 0, fro the right-had side of 3.3) ad Lea we get f )= = D f x) dx D x + ) dx + ) D Fro 3.)ad3.)weobtai + x dx. 3.) γ r ) D + = = D Siilarly, we also have f )= x dx x dx = D f x) dx D x +) dx = D +) D + + 3. 3.) x dx
You adchejoural of Iequalities ad Applicatios 08) 08:7 Page 7 of 8 ad γ r ) D + = + = D + x dx x dx = D Cobiig 3.)ad3.6) copletesthe proofof.6). Notig that r 3 ) = 0, we easily deduce where = +) 3. 3.6) r 3 ) γ = r3 ) r 3 +) ) = g), 3.7) = g)=r 3 ) r 3 +). Let D = 3.ByusigtheMatheatica software we have 88 ad g x) D x +) 6 = P x) 88 + ) 6 + )3 + ) + ) + 8 + ) + 3 )P 3 x) >0 g x) D x + )6 P x)x ) +,0,98,00,087,77 = 9x + x) + x) 6 3 + x) + x ) + 8x +x ) + x 3 )P 3 x) <0. Hece, for x, D +) 6 < g x)<d x +. 3.8) )6 Sice g )=0,by3.8)weget g)= = D g x) dx D + ) D x + ) 6 dx + It follows fro 3.7), 3.9), ad Lea that r 3 ) γ D + = = D x dx x dx = D 0 x dx. 3.9). 3.0)
You adchejoural of Iequalities ad Applicatios 08) 08:7 Page 8 of 8 Fially, ad g)= g x) dx D x +) 6 dx = D +) D r 3 ) γ D + = + = D + + + x dx x dx = D 0 x dx Cobiig 3.0)ad3.) copletestheproofof.7). +). 3.) Ackowledgeets This work was supported by the Natioal Natural Sciece Foudatio of Chia Grat Nos. 767, 60303, ad 938) ad Beijig Muicipal Coissio of Educatio Sciece ad Techology Progra KM08007009. Coputatios ade i this paper were perfored usig Matheatica 9.0. Copetig iterests The authors declare that they have o copetig iterests. Authors cotributios The authors read ad approved the fial auscript. Author details Departet of Matheatics ad Physics, Beijig Istitute of Petrocheical Techology, Beijig, P.R. Chia. Departet of Matheatics, Wuha Textile Uiversity, Wuha, P.R. Chia. Publisher s Note Spriger Nature reais eutral with regard to jurisdictioal clais i published aps ad istitutioal affiliatios. Received: 9 Jauary 08 Accepted: April 08 Refereces. Bret, R.P., Ziera, P.: Moder Coputer Arithetic. Cabridge Moographs o Applied ad Coputatioal Matheatics, vol. 8. Cabridge Uiversity Press, Cabridge 0). pp. xvi+. Dece, T.P., Dece, J.B.: A survey of Euler s costat. Math. Mag. 8, 6 009) 3. Havil, J.: Gaa: Explorig Euler s Costat. Priceto Uiversity Press, Priceto 003). Lagarias, J.C.: Euler s costat: Euler s work ad oder developets. Bull. A. Math. Soc. N.S.) 0), 7 68 03). Che, C.P., Mortici, C.: New sequece covergig towards the Euler Mascheroi costat. Coput. Math. Appl. 6, 39 398 0) 6. DeTeple, D.W.: A quicker covergece to Euler s costat. A. Math. Mo. 00),68 70 993) 7. Gavrea, I., Iva, M.: Optial rate of covergece for sequeces of a prescribed for. J. Math. Aal. Appl. 0), 3 3 03) 8. Gourdo, X., Sebah, P.: Collectio of forulae for the Euler costat. http://ubers.coputatio.free.fr/costats/gaa/gaaforulas.pdf,orsee http://ubers.coputatio.free.fr/costats/costats.htl 9. Dawei, L.: A ew quiker sequece coverget to Euler s costat. J. Nuber Theory 36, 30 39 0) 0. Dawei, L.: Soe quicker classes of sequeces coverget to Euler s costat. Appl. Math. Coput. 3, 7 77 0). Dawei, L.: Soe ew coverget sequeces ad iequalities of Euler s costat. J. Math. Aal. Appl. 9), 0). Dawei, L.: Soe ew iproved classes of covergece towards Euler s costat. Appl. Math. Coput. 3, 3 0) 3. Mortici, C.: O ew sequeces covergig towards the Euler Mascheroi costat. Coput. Math. Appl. 98), 60 6 00). Mortici, C., Che, C.P.: O the haroic uber expasio by Raauja. J. Iequal. Appl. 03 03). Mortici, C.: Product approxiatios via asyptotic itegratio. A. Math. Mo. 7),3 00) 6. Mortici, C.: New approxiatios of the gaa fuctio i ters of the digaa fuctio. Appl. Math. Lett. 3, 97 00 00)