Some new sequences that converge to the Ioachimescu constant

Transcription

1 You et al. Journal of Inequalities and Applications (06) 06:48 DOI 0.86/s x R E S E A R C H Open Access Some new sequences that converge to the Ioachimescu constant Xu You *, Di-Rong Chen,3 and Hong Shi * Correspondence: youxu@bipt.edu.cn Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 067, P.R. China Full list of author information is available at the end of the article Abstract The purpose of this paper is to give some sequences that converge quicly to the Ioachimescu constant by a multiple-correction method. MSC: Y60; A55; 4A5 Keywords: Ioachimescu constant; multiple-correction method; approximation; rate of convergence Introduction In 895, Ioachimescu (see [])introduceda constant l,which todaybearshisname, asthe limit of the sequence defined by I n =+ + + n ( n ), n N. The sequence I(n) n has attracted much attention lately and several generalizations have been given (see, e.g., [, 3]). Recently, Chen, Li and Xu [4] haveobtainedthecomplete asymptotic expansion of the Ioachimescu sequence, I n l + n b (4 3)!!, n N, ()! n / where b n denotes the nth Bernoulli number. One easily obtains the following representations of the Ioachimescu constant: x + x l = dx 0 ( + x) 3/ and l = ( + ). 06 You et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a lin to the Creative Commons license, and indicate if changes were made.

2 You et al. Journal of Inequalities and Applications (06) 06:48 Page of 6 A representation of the Ioachimescu constant has also been given by Ramanujan (95) [5], l = ( +) ( ) +. From it one easily obtains a representation of the Ioachimescu constant in terms of the extended ζ function l = ζ ( ) +. As a result of [], we have l = Let a (0, + )ands (0, ), the sequence y n (a, s)= a + s (a +) + + s (a + n ) [ (a + n ) s a s], n N, s s is convergent [3] and its limit is a generalized Euler constant denoted by l(a, s). Clearly, l(, /) = l. Furthermore, Sîntămărian has proved that lim ns( y n (a, s) l(a, s) ) =. Alsoin [3], considering the sequence u n (a, s)=y n (a, s) she has proved that (a + n ) s, lim ns+( l(a, s) u n (a, s) ) = s and, for the sequence α n (a, s)= a + s (a +) + + s (a + n ) (( a + n ) s ) a s, n N, s s she has proved that lim ns+( α n (a, s) l(a, s) ) = s 4. In [6, 7], Sîntămărian has obtained some new sequences that convergence to l(a, s) with therateofconvergencen s 5. Other results regarding l(a, s) canbefoundin[8 0] and some of the references therein. In our paper, we will give some sequences that converge quicly to the Ioachimescu constant l by a multiple-correction method [ 3], based on the sequence I(n)=+ + + n ( n ), n N.

3 You et al. Journal of Inequalities and Applications (06) 06:48 Page 3 of 6 Sequences convergent to the Ioachimescu constant l The following lemma gives a method for measuring the rate of convergence; for its proof see Mortici [4, 5]. Lemma If the sequence (x n ) n N is convergent to zero and there exists the limit lim n + ns (x n x n+ )=l [,+ ], (.) with s >,then lim n + ns x n = l s. (.) Now we apply multiple-correction method to study faster convergence sequences for the Ioachimescu constant, and this method could be used to solve other problems, such as the Euler-Mascheroni constant, Glaisher-Kinelin s and Bendersy-Adamchi s constants, Somos quadratic recurrence constant, and so on [6 9]. Theorem For the Ioachimescu constant, we have the following convergent sequence: where I () i (n)= ( n )+η () 0 (n)+η() (n)+ + η() i (n), (.3) η () 0 (n)=0, η() (n)= n + 6 η (), (.4) (n)= 9,... (.5) n n n3 + 7,833 46,656 n + 76,647 6,873,856 n 9,96,4 806,5,680 Proof (Step ) The initial correction. We choose η () 0 (n)=0,andlet I () 0 (n):=i(n)+η() 0 (n)= ( n )+η () (n). (.6) Developing equation (.6) into a power series expansion in /n,we have I () 0 (n) I() 0 (n +)= 4 n 3 0 ( ) + O. (.7) n 5 By Lemma,therateofconvergenceof(I () 0 (n) l) n N is n,since lim n ( () 0 (n) l) =. (Step)Thefirstcorrection.Let η () (n)= a n + b0 (.8)

4 You et al. Journal of Inequalities and Applications (06) 06:48 Page 4 of 6 and define I () (n):= ( n )+η () 0 (n)+η() (n). (.9) Developing (.9)intopowerseriesexpansionin/n,weobtain I () (n) I() (n +)= a + 4 n 3 + 3a ( + b 0 ) ( ( 3+4a +3b0 +3b 64 0)) n 7 n 5 ( + O n 9 ). (.0) (i) If a, the rate of convergence of the (I() (n) l) n N is n,since lim n ( () (n) l) = a (ii) If a = and b 0 =,from(.0) we obtain 6 I () 5 (n) I() (n +)= 384 n 7 ( ) + O. n 9 Then the rate of convergence of the (I () (n) l) n N is n 5,since lim n 5 ( () (n) l) = 9. (Step 3) The second correction. Similarly, set the second-correction function η () (n)= a (.) n5 + b 4 n 4 + b 3 n 3 + b n + b n + b 0 and define I () (n):= ( n )+η () 0 (n)+η() (n)+η() (n). (.) Bythesamemethodasabove,wegeta = 9, b 4 = 3 8, b 3 = 34 88, b = 7,833 46,656, b = 76,647 6,873,856, b 0 = 9,96,4 806,5,680. Applying Lemma again, one has () (n) I() (n +)) =,87,793,943,49 67,483,03,447,680, (.3) () (n) l) 75,75,584,897 = 33,74,506,73,840. (.4) Repeating the above approach for the Ioachimescu constant, we can prove Theorem.

5 You et al. Journal of Inequalities and Applications (06) 06:48 Page 5 of 6 3 Other sequences convergent to the Ioachimescu constant l In this section, we provide some other approximation for the Ioachimescu constants by a multiple-correction method. The initial correction is the same as above, we change the correction function from step. (Step ) The first-correction. Let the second-correction function be η () (n)= a n + u n+v (3.) and define I () (n):= ( a n )+. (3.) n + u n+v Bythesamemethodasabove,wefinda =, u = 6, v = 8. Applying Lemma,onehas () (n) I () (n +) ) = 59 7,648, (3.3) () (n) l ) 37 = 3,84. (3.4) Repeating the above approach for the Ioachimescu constant, we can prove the following theorem. Theorem For the Ioachimescu constant, we have the following convergent sequence: I i (n)= ( n )+ n + a u u n+v + u n+v + 3 n+v u i n+v i, (3.5) where a =, u = 6, v = 8 ; u = , v = 888 ; (3.6) u 3 = 837,738, v 3 = 33 8,35 ; (3.7) u 4 =,3,79 3,690,40, v 4 = 6,349 9,950,896 ; u 5 = 393,86,357,59 35,9,348,00, (3.8) v 5 = 5,0,056,744,79 ;,70,864,635,038,456. (3.9) Remar Theorem provides some quasi-continued fraction sequences with a faster rate of convergence for the Ioachimescu constant. Competing interests The authors declare that they have no competing interests.

6 You et al. Journal of Inequalities and Applications (06) 06:48 Page 6 of 6 Authors contributions The authors read and approved the final manuscript. Author details Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 067, P.R. China. Department of Mathematics, Wuhan Textile University, Hubei Wuhan, 43000, P.R. China. 3 School of Mathematics and System Science, Beihang University, Beijing, 009, P.R. China. Acnowledgements We are grateful to the editor and anonymous reviewers for their valuable comments and corrections that helped improve the original version of this paper. The research was supported by the National Natural Science Foundation of China under grant no , 5767, and Received: 8 March 06 Accepted: 5 May 06 References. Ioachimescu, AG: Problem 6. Gaz. Math. (), 39 (895). Sîntămărian, A: Some inequalities regarding a generalization of Ioachimescu s constant. J. Math. Inequal. 4(3), (00) 3. Sîntămărian, A: Regarding a generalisation of Ioachimescu s constant. Math. Gaz. 94(530),70-83 (00) 4. Chen, CP, Li, L, Xu, YQ: Ioachimescu s constant. Proc. Jangjeon Math. Soc. 3, (00) 5. Ramanujan, S: On the sum of the square roots of the first n natural numbers. J. Indian Math. Soc. 7, (95) 6. Sîntămărian, A: Sequences that converge quicly to a generalized Euler constant. Math. Comput. Model. 53, (0) 7. Sîntămărian, A: Some new sequences that converge to a generalized Euler constant. Appl. Math. Lett. 5, (0) 8. Sîntămărian, A: A generalisation of Ioachimescu s constant. Math. Gaz. 93(58), (009) 9. Sîntămărian, A: Some sequences that converge to a generalization of ioachimescu s constant. Autom. Comput. Appl. Math. 8(), (009) 0. Sîntămărian, A: Sequences that converge to a generalization of Ioachimescu s constant. Sci. Stud. Res. Ser. Math. Inform. 0(), (00). Cao, XD, Xu, HM, You, X: Multiple-correction and faster approximation. J. Number Theory 49, (05). Cao, XD: Multiple-correction and continued fraction approximation. J. Math. Anal. Appl. 44, (05) 3. Cao, XD, You, X: Multiple-correction and continued fraction approximation (II). Appl. Math. Comput. 6, 9-05 (05) 4. Mortici, C: On new sequences converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 59(8), (00) 5. Mortici, C: Product approximations via asymptotic integration. Am. Math. Mon. 7(5), (00) 6. Xu, HM, You, X: Continued fraction inequalities for the Euler-Mashcheroni constant. J. Inequal. Appl. 04,343 (04) 7. You, X: Some new quicer convergences to Glaisher-Kinelin s and Bendersy-Adamchi s constants. Appl. Math. Comput. 7, 3-30 (05) 8. You, X, Chen, D-R: Improved continued fraction sequence convergent to the Somos quadratic recurrence constant. J. Math. Anal. Appl. 436, (06) 9. You, X, Huang, SY, Chen, D-R: Some new continued fraction sequence convergent to the Somos quadratic recurrence constant. J. Inequal. Appl. 06, 9 (06)