#A11 INTEGERS 11 (2011) NEW SEQUENCES THAT CONVERGE TO A GENERALIZATION OF EULER S CONSTANT

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1 #A INTEGERS (20) NEW SEQUENCES THAT CONVERGE TO A GENERALIZATION OF EULER S CONSTANT Alin Sîntămărin Deprtment of Mthemtics, Technicl University of Cluj-Npoc, Cluj-Npoc, Romni Alin.Sintmrin@mth.utcluj.ro Received: 7/3/0, Revised: 0/28/0, Accepted: 2/24/0, Published: /3/ Abstrct The purpose of the pper is to give some sequences tht converge quickly to generliztion of Euler s constnt, i.e., the it of the sequence where (0, ). n ln n, n N. Introduction Euler s constnt, being one of the most importnt constnts in mthemtics, ws investigted by mny mthemticins. Usully denoted by γ, this constnt is the it of the sequence (D n ) n N defined by D n = 2 n ln n, for ech n N. It is well-known tht n(d n γ) = n 2 (see [], [2], [3], [5, pp ], [7], [3, Problem 8, pp. 38, 97], [4], [2], [23], [24], [25], [26]). In order to increse the slow rte of convergence of the sequence (D n ) n N to γ, D. W. DeTemple considered in [4] the sequence (R n ) n N defined by R n = 2 n ln n 2, for ech n N, nd he proved tht 24(n) < R 2 n γ < 24n, 2 for ech n N. L. Tóth used in [22] the sequence (T n ) n N defined by T n = 2 n ln n 2 24n, for ech n N, nd T. Negoi proved in [2] tht 48(n) < 3 γ T n < 48n, for ech n N. 3 Let (0, ). We consider the sequence (y n ()) n N defined by y n () = n ln n, for ech n N. The sequence (y n ()) n N is convergent (see, for exmple, [6, p. 453]; see lso [5], [6], [7], [8], [9], [20] nd some of the references therein) nd

2 INTEGERS: (20) 2 its it, denoted by γ(), is generliztion of Euler s constnt. Clerly, γ() = γ. Numerous results regrding the generliztion of Euler s constnt γ() we hve obtined in [5], [6], [7], [8], [9] nd [20]. We mention the following representtion of γ() ([9, Theorem 2.2.4, p. 78]): m γ() = y n 2( n ) k= B (2m)! ( n ) n P 2m (x) dx, ( x ) 2m2 for ech n N, ny m N, where B is the Bernoulli number of index nd P 2m (x) = () m k= 2 sin(πx) (π) 2m, obtined by pplying the Euler-Mclurin summtion formul ([6, p. 524], [5, p. 86]). If we tke = in the bove-mentioned representtion, then we obtin result presented, for exmple, in [6, pp. 527, 528], [5, pp. 88, 89]. Recent results regrding Euler s constnt hve been obtined by C. Mortici in [9], [0], []. Also, we remind the following lemm (C. Mortici [8, Lemm]), which is consequence of the the Stolz-Cesro Theorem, the cse 0 0. Lemm. Let (x n ) n N be convergent sequence of rel numbers nd x = n x n. We suppose tht there exists α R, α >, such tht Then there exists the it n nα (x n x n ) = l R. n nα (x n x ) = l α. In Section 2 we present clsses of sequences with the rgument of the logrithmic term modified nd tht converge quickly to γ(). 2. Sequences Tht Converge to γ() Theorem 2. Let (0, ). We specify tht γ() is the it of the sequence (y n ()) n N from Introduction. (i) We consider the sequence (α n,2 ()) n N defined by α n,2 () = n 2( n ) 2( n ) 2 n ln 20( n ) 3,

3 INTEGERS: (20) 3 for ech n N. Then n n6 (γ() α n,2 ()) = 252. (ii) We consider the sequence (β n,2 ()) n N defined by for ech n N. Then Proof. (i) We hve β n,2 () = α n,2 () 252( n ) 6, n n8 (β n,2 () γ()) = α n,2 () α n,2 () = 2( n) 2( n ) 2( n) 2 2( n ) 2 ln n ln n 20( n) 3 20( n ) 3 = 2( n) 2( n) 2( n) 2 2 2( n) 2 ln 20( n) 4 ln n 3, 20( n) 4 for ech n N. Set ε n :=, for ech n N. Since ε n (, ), 20 ε4 n (, ] nd ε n 20 ε 4 n (ε n) (, ], for ech n N \ {}, using the series expnsion 3 ([6, pp. 7 79, p. 209]) we obtin α n,2 () α n,2 () ε n = 2 ε n 2 ε n 2 ε2 n 2 ln 20 ε4 n ln ε 2 n ( ε n ) 2 ε n 20 ε 4 n ( ε n ) 3 = 42 ε7 n 2 ε8 n ε9 n ε0 n O(ε n ), for ech n N \ {}. It follows tht n n7 (α n,2 () α n,2 ()) = 42.

4 INTEGERS: (20) 4 Now, ccording to Lemm, we get (ii) We re ble to write tht β n,2 () β n,2 () = α n,2 () α n,2 () = α n,2 () α n,2 () n n6 (γ() α n,2 ()) = ( n ) 6 252( n) 6 252( n) 6 6 = α n,2 () α n,2 () 252 ε 6 n ( ε n ) ε6 n = ε9 n ε0 n O(ε n ), for ech n N \ {}. It follows tht Now, ccording to Lemm, we get n n9 (β n,2 () β n,2 ()) = n n8 (β n,2 () γ()) = ( n) 6 In the sme mnner s in the proof of Theorem 2, considering the sequence in ech of the following prts, we get the indicted it: δ n,2 () = β n,2 () () 8, for ech n N, n n0 (γ() δ n,2 ()) = 32 ; η n,2 () = δ n,2 () 32(), for ech n N, 0 n n2 (η n,2 () γ()) = ; θ n,2 () = η n,2 () (), for ech n N, 2 n n4 (γ() θ n,2 ()) = 2 ; λ n,2 () = θ n,2 () 2(), for ech n N, 4 n n6 (λ n,2 () γ()) = ; µ n,2 () = λ n,2 () (), for ech n N, 6 n n8 (γ() µ n,2 ()) =

5 INTEGERS: (20) 5 We point out the pttern in forming the sequences from Theorem 2 nd those mentioned bove. For exmple, the generl term of the sequence (µ n,2 ()) n N cn be written in the form with µ n,2 () = n 2( n ) B 2 2 n 8 ln B 4 4 ( n ) 3 k=3 B, if k = 2p, p N, c k,2 = B 2 k B4 2, if k = 2p 2, p N, k 4 ( n ) 2 c k,2 ( n ), where B is the Bernoulli number of index. Relted to this remrk, see lso [6, Remrk 3.4], [9, p. 7, Remrk 2..3; pp. 00, 0, Remrk 3..6]. For Euler s constnt γ = we obtin, for exmple: α 2,2 () = ; α 3,2 () = ; β 2,2 () = ; β 3,2 () = ; δ 2,2 () = ; δ 3,2 () = ; η 2,2 () = ; η 3,2 () = ; θ 2,2 () = ; θ 3,2 () = ; λ 2,2 () = ; λ 3,2 () = ; µ 2,2 () = ; µ 3,2 () = As cn be seen, µ 3,2 () is ccurte to eight deciml plces in pproximting γ. Theorem 3. Let (0, ). We specify tht γ() is the it of the sequence (y n ()) n N from Introduction. (i) We consider the sequence (α n,3 ()) n 2 defined by α n,3 () = n 2( n ) 2( n ) 2 20( n ) 4 n ln for ech n N \ {}. Then 252( n ) 5 n n8 (α n,3 () γ()) = 240.,

6 INTEGERS: (20) 6 (ii) We consider the sequence (β n,3 ()) n 2 defined by for ech n N \ {}. Then β n,3 () = α n,3 () 240( n ) 8, n n0 (γ() β n,3 ()) = 32. (iii) We consider the sequence (δ n,3 ()) n 2 defined by for ech n N \ {}. Then Proof. (i) We hve δ n,3 () = β n,3 () 32( n ) 0, n n2 (δ n,3 () γ()) = α n,3 () α n,3 () = 2( n ) 2( n) 2( n ) 2 2( n) 2 20( n ) 4 20( n) 4 ln n ln n = 2( n) 20( n) 4 252( n) 5 2( n) 2 2( n) 2 252( n ) ( n) 4 ln n 252( n) 6 5 ln 2( n) 2 252( n) 6, for ech n N \ {}. Set ε n :=, for ech n N \ {}. Since ε n (, ), ε n 252 ε 6 n (ε n) (, ] nd ε6 n (, ], for ech n N \ {}, using the

7 INTEGERS: (20) 7 series expnsion ([6, pp. 7 79, p. 209]) we obtin α n,3 () α n,3 () = 2 ε n ε n 2 ε n 2 ε 2 n ( ε n ) 2 2 ε2 n 20 ε 4 n ( ε n ) 4 20 ε4 n ln ε n 252 ε 6 n ( ε n ) 5 ln 252 ε6 n = 30 ε9 n 3 20 ε0 n 4 33 ε n ε2 n ε3 n ε4 n O(ε 5 n ), for ech n N \ {}. It follows tht Now, ccording to Lemm, we get (ii) We re ble to write tht n n9 (α n,3 () α n,3 ()) = 30. n n8 (α n,3 () γ()) = 240. β n,3 () β n,3 () = α n,3 () α n,3 () 240( n) 8 240( n ) 8 = α n,3 () α n,3 () 240( n) ( n) 8 = α n,3 () α n,3 () 240 ε8 n 240 ε 8 n ( ε n ) 8 = 5 66 ε n for ech n N \ {}. It follows tht 5 2 ε2 n ε3 n ε4 n O(ε 5 n ), Now, ccording to Lemm, we get n n (β n,3 () β n,3 ()) = n n0 (γ() β n,3 ()) = 32.

8 INTEGERS: (20) 8 (iii) We hve δ n,3 () δ n,3 () = β n,3 () β n,3 () = β n,3 () β n,3 () 32( n ) 0 32( n) 0 32( n) 0 0 = β n,3 () β n,3 () 32 ε 0 n ( ε n ) 0 32 ε0 n = ε3 n ε4 n O(ε 5 n ), for ech n N \ {}. It follows tht Now, ccording to Lemm, we get n n3 (δ n,3 () δ n,3 ()) = n n2 (δ n,3 () γ()) = ( n) 0 In the sme mnner s in the proof of Theorem 3, considering the sequence in ech of the following prts, we get the indicted it: η n,3 () = δ n,3 () () 2, for ech n N \ {}, n n4 (γ() η n,3 ()) = 2 ; θ n,3 () = η n,3 () 2(), for ech n N \ {}, 4 n n6 (θ n,3 () γ()) = ; 367 λ n,3 () = θ n,3 () 860(), for ech n N \ {}, 6 n n8 (γ() λ n,3 ()) = ; µ n,3 () = λ n,3 () (), for ech n N \ {}, 8 n n20 (µ n,3 () γ()) = We point out the pttern in forming the sequences from Theorem 3 nd those mentioned bove. For exmple, the generl term of the sequence (µ n,3 ()) n 2 cn

9 INTEGERS: (20) 9 be written in the form with µ n,3 () = n 2( n ) B 2 2 ( n ) 2 B 4 4 ( n ) 4 n ln B 6 6 ( n ) 5 9 k=4 B, if k = 3p, p N, B c k,3 =, if k = 3p 2, p N, B 3 k B6 6 k 3, if k = 3p 3, p N, c k,3 ( n ), where B is the Bernoulli number of index. Relted to this remrk, see lso [6, Remrk 3.4], [9, p. 7, Remrk 2..3; pp. 00, 0, Remrk 3..6]. For Euler s constnt γ = we obtin, for exmple: α 2,3 () = ; α 3,3 () = ; β 2,3 () = ; β 3,3 () = ; δ 2,3 () = ; δ 3,3 () = ; η 2,3 () = ; η 3,3 () = ; θ 2,3 () = ; θ 3,3 () = ; λ 2,3 () = ; λ 3,3 () = ; µ 2,3 () = ; µ 3,3 () = As cn be seen, λ 3,3 () nd µ 3,3 () re ccurte to eight deciml plces in pproximting γ. Concluding, the following remrk cn be mde. Let (0, ) nd q N \ {}. Let n 0 = min n N n B2q 2q () > 0. We consider the 2q sequence (α n,q ()) n n0 defined by α n,q () = n n ln B 2q 2q q 2( n ) ( n ) 2q, k= B ( n )

10 INTEGERS: (20) 0 for ech n N, n n 0. Clerly, α n,q() = γ(). Bsed on the sequence n (α n,q ()) n n0, clss of sequences convergent to γ() cn be considered, nmely where {(α n,q,r ()) n n0 r N, r q }, α n,q,r () = α n,q () r k=q c k,q ( n ), for ech n N, n n 0, with B, if k {qp, qp 2,..., qp q }, p N, c k,q = B q k B2q q, if k = qp q, p N. k 2q In this section we hve obtined tht the sequence (α n,2 ()) n N converges to γ() with order 6 nd tht the sequence (α n,3 ()) n 2 converges to γ() with order 8. We hve lso obtined tht the sequence (α n,2,r ()) n N converges to γ() with order 2(r ), for r {3, 4, 5, 6, 7, 8}, nd tht the sequence (α n,3,r ()) n 2 converges to γ() with order 2(r ), for r {4, 5, 6, 7, 8, 9}. References [] H. Alzer, Inequlities for the gmm nd polygmm functions, Abh. Mth. Semin. Univ. Hmb. 68, 998, [2] R. P. Bos, Estimting reminders, Mth. Mg. 5 (2), 978, [3] C.-P. Chen, F. Qi, The best lower nd upper bounds of hrmonic sequence, RGMIA 6 (2), 2003, [4] D. W. DeTemple, A quicker convergence to Euler s constnt, Amer. Mth. Monthly 00 (5), 993, [5] J. Hvil, Gmm. Exploring Euler s Constnt, Princeton University Press, Princeton nd Oxford, [6] K. Knopp, Theory nd Appliction of Infinite Series, Blckie & Son Limited, London nd Glsgow, 95. [7] S. K. Lkshmn Ro, On the sequence for Euler s constnt, Amer. Mth. Monthly 63 (8), 956, [8] C. Mortici, New pproximtions of the gmm function in terms of the digmm function, Appl. Mth. Lett. 23 (), 200, [9] C. Mortici, Improved convergence towrds generlized Euler-Mscheroni constnt, Appl. Mth. Comput. 25 (9), 200,

11 INTEGERS: (20) [0] C. Mortici, On new sequences converging towrds the Euler-Mscheroni constnt, Comput. Mth. Appl. 59 (8), 200, [] C. Mortici, On some Euler-Mscheroni type sequences, Comput. Mth. Appl. 60 (7), 200, [2] T. Negoi, O convergenţă mi rpidă către constnt lui Euler (A quicker convergence to Euler s constnt), Gz. Mt. Seri A 5 (94) (2), 997, 3. [3] G. Póly, G. Szegö, Aufgben und Lehrsätze us der Anlysis (Theorems nd Problems in Anlysis), Verlg von Julius Springer, Berlin, 925. [4] P. J. Rippon, Convergence with pictures, Amer. Mth. Monthly 93 (6), 986, [5] A. Sîntămărin, About generliztion of Euler s constnt, Automt. Comput. Appl. Mth. 6 (), 2007, [6] A. Sîntămărin, A generliztion of Euler s constnt, Numer. Algorithms 46 (2), 2007, 4 5. [7] A. Sîntămărin, Some inequlities regrding generliztion of Euler s constnt, J. Inequl. Pure Appl. Mth. 9 (2), 2008, 7 pp., Article 46. [8] A. Sîntămărin, A representtion nd sequence trnsformtion regrding generliztion of Euler s constnt, Automt. Comput. Appl. Mth. 7 (2), 2008, [9] A. Sîntămărin, A Generliztion of Euler s Constnt, Editur Medimir, Cluj-Npoc, [20] A. Sîntămărin, Approximtions for generliztion of Euler s constnt, Gz. Mt. Seri A 27 (06) (4), 2009, [2] S. R. Tims, J. A. Tyrrell, Approximte evlution of Euler s constnt, Mth. Gz. 55 (39), 97, [22] L. Tóth, Asupr problemei C: 608 (On problem C: 608), Gz. Mt. Seri B 94 (8), 989, [23] L. Tóth, Problem E 3432, Amer. Mth. Monthly 98 (3), 99, 264. [24] L. Tóth, Problem E 3432 (Solution), Amer. Mth. Monthly 99 (7), 992, [25] A. Vernescu, Ordinul de convergenţă l şirului de definiţie l constntei lui Euler (The convergence order of the definition sequence of Euler s constnt), Gz. Mt. Seri B 88 (0-), 983, [26] R. M. Young, Euler s constnt, Mth. Gz. 75 (472), 99,