NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE

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1 UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece to the costat γ Our results exted, improve ad uify some existig results i this directio Keywords: Euler-Mascheroi costat, speed of covergece, asymptotic expasio MSC200: B4, 26A48, 4A60, 4A25 Itroductio Oe of the most kow costat i mathematics is the Euler-Mascheroi costat γ = , which is defied as the it of sequece γ = log ) This sequece has diverse applicatios i may areas of mathematics, ragig from classical or umerical aalysis to umber theory, special fuctios or theory of probability There is a huge literature about the sequece γ ) ad the costat γ Please refer to [, 4, 5, 6, 7, 8] ad all the refereces therei The speed of covergece to γ of sequece γ ) is very slowly, if we take ito accout that it coverges like That is why may authors develop studies to improve the speed of covergece of sequece γ ) For example, Cesaro [] proved that for every positive iteger, there exists a umber c 0, ) such that the followig relatio is true: c 2 log2 + ) γ = 6 + ) Recetly, by chagig the logarithmic term i ), Che ad Li [2] itroduced the sequeces P = log ) ad Q = log [ ad proved that the followig iequalities hold: 80 + ) 4 < γ P < 80 4, ad ) 6 < Q 8 γ < 285 6, ) ] 2 45 Departmet of Mathematics ad Computer Sciece, Uiversity Duărea de Jos, Domească Street, No 47, Galaţi, , Romaia, gbercu@ugalro 5

2 6 Gabriel Bercu for all itegers, I sectio 2, we itroduce the sequece ω = r logr + b r ), 2) where r ad b are positive real costats Our aim is to fid values for r ad b which provide a faster covergece of the sequece ω ) to the Euler-Mascheroi costat γ I sectio sectio, we discuss o the faster covergece towards the costat γ of a sequece with logarithmic term ivolvig the costat e I this part, we make a lik betwee our study ad the research work of Mortici [5] 2 A ew fast coverget sequece to the costat γ A importat tool for computig the speed of the covergece is Stolz lemma, the case 0/0 Our study is based o a variat of this lemma Lemma 2 If ω ) is coverget to zero ad there exists the it the there exists the it k ω + ω ) = l R, k >, k ω = l k At first, we calculate the differece ω + ω After some calculatio, we fid ω + ω = + r log + ) [ log + b + ) log + b )] r r The, we use a computer software to write the expressio ω + ω as power series of Thus ω + ω = 2b r 2 + b2 b r + 4b + 6b 2 ) + 4b r 4r ) The best speed of covergece of the sequece ω ) is obtaied i case whe first coefficiets of ) vaish: 2b r = 0, b 2 b r = 0 We fid b = ad r = 2 I this case, the coefficiet of 4 becomes 8 We ca state the followig Theorem 2 The followig statemets hold true: i) If 2b r 0, the the speed of covergece of sequece ω ) is, sice 2 ω + ω ) = 2b r, ad ω γ) = r 2b 0 ii) If 2b r = 0, ad b 2 b 0, that is b, the the speed of covergece of the sequece ω ) is 2, sice ω + ω ) = b2 b, ad r 2 ω γ) = + b2 + b 0 6r iii) If 2b r = 0, ad b 2 b = 0, that is b = ad r = 2, the the speed of covergece of the sequece ω ) is, sice 4 ω + ω ) =, ad 8 ω γ) = 54

3 New fast coverget sequeces of Euler-Mascheroi type 7 Remark 2 For b = ad r = 2, the sequece itroduced by us has the form ω 0 = ) If deote by ψ the digamma fuctio, it is well kow that ψ + ) = γ + k 4) By the elegat article by Mortici ad Che [8], we obtai x 2x 2 + 6x 0x 5 < ψ x + ) < x 2x 2 +, 6x x > 0 5) We will prove the followig result Theorem 22 If ω) 0 is defied as above, the the followig relatios hold: 54 + ) < γ ω0 < 54, for ay iteger, Proof First of all, we observe that γ ω 0 54 = γ k ) 54 ad, by 4), k= k= γ ω 0 54 = 2 + ) ψ + ) 54 6) This eable us to cosider the fuctio F : 0, ) R, F x) = x2 + ) ψx + ) x 54x, whose formula ca be writte i a more coveiet form as F x) = logx + ) log ) 2 log x ψx + ) 54x Now, after some calculatio ad usig 5), we get F x) > 8x 4 > 0, x > 0 x + ) Therefore, the sequece F )) is icreasig, hece From 6), it follows that To prove that F ) < F ) = 0 γ ω 0 54 < 0 γ ω ) > 0, we cosider the fuctio G: [, ) R, Gx) = x2 + ) ψx + ) x 54x + ) For calculatio, we use a more coveiet form of G as Gx) = 2 logx + ) 2 log x 2 log ψx + ) 54x + )

4 8 Gabriel Bercu Usig the same techique as above, we fid that G x) < 0, that the sequece G) is decreasig, hece x > This meas G) > G) = 0, ad this completes the proof I the followig we itroduce the sequece R ), R = r logr + b r + c r 2 ), 7) r, b ad c beig positive real costats Note that for c = 0 we fid the sequece ω ), itroduced i 2) We also calculate the differece R + R = + r 2 log r + ) r [ log + b b + c + ) 2 log + b + c )] 2 Usig a computer software to write the expressio R + R as power series of, we obtai R + R = 2b r 2 + b2 b + 6c r + r 2bc + 4b + 6b 2 + 4b 2c 4r 4 + 4r + 2 5b + 2)b + c + )2 + 5bc 2 + 5b + 2) b + c + ) 5b c + b 5 b + 2) 5 ) 5r +0 6 We vaish the first three coefficiets, ad fid 2b r = 0 b 2 b + 6c = 0 r r 2bc + 4b 2c + 4b + 6b 2 = 0 4r 5 The solutio of this system is b =, c = ad r = 2 We also have 5 R + R ) = ad usig Lemma 2, we obtai 45 4 R γ) = Therefore, by our method, we obtai that the sequece 80 has the speed of covergece 4 R = log ) Remark 22 Note that the sequece R ) is the sequece P ) itroduced by Che ad Li i [2] Therefore, we proved that this oe is the uique sequece of the form 7), which has the speed of covergece 4

5 New fast coverget sequeces of Euler-Mascheroi type 9 A ew fast coverget sequece with logarithmic term ivolvig the costat e Our aim i this sectio is to discuss o the faster covergece towards the costat γ of a sequece with logarithmic term ivolvig the costat e I this respect, we shall refer as startig poit the work of Mortici [5] I this research article, he itroduced ad studied the speed of covergece to γ for sequeces of the form µ = log expa/ + b)) ) log a Adaptig our iitial sequece ω ) for r = 2, we defie a ew sequece µ = expa/ 2 ) 2 log + b)) a As above, we write µ + µ = log log + ) expa/ + ) + b + )) a +)+b+) + log + b + ) 2 log ) log + b ) ) expa/ + b))) a +b) Now, we use a computer software to obtai the followig represetatio i power series: µ + µ = b b2 + b + 2 a ) b + b 2 + 2b + a ab ) 8) b 5 b + ) 5 + a 0 2 ab ab2 a2 2 We cacel the first coefficiets of 8), b =, b2 + b a 2 = 0 ) ) Solvig with respect to a ad b, we fid that the solutio is a = 2 ad b = I this case, the coefficiet of 4 is also 0 ad the coefficiet of 5 is 5 Therefore, we have prove this result Theorem The followig statemets hold good: i) If b, the the speed of covergece of sequece µ ) is, sice sice sice 2 µ + µ ) = b, ad 2 µ γ) = b 2 ii) If b = ad a 2, the the speed of covergece of sequece µ ) is 2, µ + µ ) = a, ad 2 2 µ γ) = a ) iii) If b = ad a = 2, the the speed of covergece of sequece µ ) is 4, 5 µ + µ ) =, ad 5 4 µ γ) = 540 )

6 40 Gabriel Bercu Remark For b = ad a = 2, the sequece µ ) has the form µ 0 = )) exp 2/ + )) 2 log We otice that the sequece µ 0 ) has the logarithm term ivolvig the costat e Remark 2 The ew sequece µ 0 ) coverges to γ like 4, while the sequece of Mortici itroduced i [5] coverges to γ like We deduce that the approximatio γ µ 0 is more accurate tha γ µ Usig a similar techique as i the proof of Theorem 22, we fid Theorem 2 If µ 0 ) is defied as above, the the followig relatios hold: < µ0 γ < , for ay iteger, R E F E R E N C E S [] Cesàro, E: Sur la serie harmoique, Nouvelles A Math 4, ) [2] Che, C-P, Li, L: Two accelerated approximatios to the Euler-Mascheroi costat, Sci Maga 6, ) [] Che, C-P, Mortici, C: New sequece covergig towards the Euler-Mascheroi costat, Comput Math Appl 64, ) [4] Mortici, C: O some Euler-Mascheroi type sequeces, Comput Math Appl 60, ) [5] Mortici, C: A quicker covergece toward the gamma costat with the logarithm term ivolvig the costat e, Carpathia J Math 26), ) [6] Mortici, C: Improved covergece towards geeralized Euler-Mascheroi costat, Appl Math Comput 25, ) [7] Mortici, C: Fast covergeces towards Euler-Mascheroi costat, Comput Math Appl 29), ) [8] Mortici, C, Che, C-P: O the harmoic umber expasio by Ramaja, J Iequal Appl ID: 20:222 20) 2