On the harmonic number expansion by Ramanujan
© Mortici and Chen; licensee Springer 2013
Received: 9 December 2012
Accepted: 12 April 2013
Published: 3 May 2013
Let denote the Euler-Mascheroni constant, and let the sequences
The main aim of this paper is to find the values r, s, t, a, b, c and d which provide the fastest sequences and approximating the Euler-Mascheroni constant. Also, we give the upper and lower bounds for in terms of .
MSC: 11Y60, 40A05, 33B15.
KeywordsEuler-Mascheroni constant harmonic numbers inequality psi function polygamma functions asymptotic expansion
Entry 9 of Chapter 38 of Berndt’s edition of Ramanujan’s Notebooks [, p.521] reads,
For the history and the development of Ramanujan’s formula, see .
and posed the following natural question.
is the sequence which would converge to γ in the fastest way.
Very recently, Yang  published the solution of the open problem (1.6) by using logarithmic-type Bell polynomials.
respectively. Our Theorems 1 and 2 are to find the values r, s, t, a, b, c and d which provide the fastest sequences and approximating the Euler-Mascheroni constant.
The speed of convergence of the sequence is .
The speed of convergence of the sequence is .
Our Theorems 3 and 4 establish the bounds for in terms of .
Remark 1 The inequality (1.14) is sharper than (1.8), while the inequality (1.13) is sharper than (1.14).
Before we prove the main theorems, let us give some preliminary results.
are called the polygamma functions.
Lemma 1 gives a method for measuring the speed of convergence.
Lemma 2 [, Theorem 9]
3 Proofs of Theorems 1-4
By using Lemma 1, we obtain the assertion of Theorem 1. □
By using Lemma 1, we obtain the assertion of Theorem 2. □
Therefore, for .
by using the asymptotic formula (2.3). This completes the proof of the second inequality in (1.13). □
Therefore, for .
by using the asymptotic formula (2.3). This completes the proof of the first inequality in (1.14). □
Remark 2 Some calculations in this work were performed by using the Maple software for symbolic calculations.
Remark 3 The work of the first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.
Dedicated to Professor Hari M Srivastava.
- Alzer H: Inequalities for the gamma and polygamma functions. Abh. Math. Semin. Univ. Hamb. 1998, 68: 363–372. 10.1007/BF02942573MathSciNetView ArticleGoogle Scholar
- Anderson GD, Barnard RW, Richards KC, Vamanamurthy MK, Vuorinen M: Inequalities for zero-balanced hypergeometric functions. Trans. Am. Math. Soc. 1995, 347: 1713–1723. 10.1090/S0002-9947-1995-1264800-3MathSciNetView ArticleGoogle Scholar
- Rippon PJ: Convergence with pictures. Am. Math. Mon. 1986, 93: 476–478. 10.2307/2323478MathSciNetView ArticleGoogle Scholar
- Tims SR, Tyrrell JA: Approximate evaluation of Euler’s constant. Math. Gaz. 1971, 55: 65–67. 10.2307/3613323MathSciNetView ArticleGoogle Scholar
- Tóth L: Problem E3432. Am. Math. Mon. 1991., 98: Article ID 264Google Scholar
- Tóth L: Problem E3432 (solution). Am. Math. Mon. 1992, 99: 684–685.View ArticleGoogle Scholar
- Young RM: Euler’s constant. Math. Gaz. 1991, 75: 187–190. 10.2307/3620251View ArticleGoogle Scholar
- Cesàro E: Sur la serie harmonique. Nouvelles Ann. Math. 1885, 4: 295–296.Google Scholar
- Chen C-P: The best bounds in Vernescu’s inequalities for the Euler’s constant. RGMIA Res. Rep. Coll. 2009., 12: Article ID 11. Available online at http://ajmaa.org/RGMIA/v12n3.phpGoogle Scholar
- Chen C-P: Inequalities and monotonicity properties for some special functions. J. Math. Inequal. 2009, 3: 79–91.MathSciNetView ArticleGoogle Scholar
- Chen C-P: Inequalities for the Euler-Mascheroni constant. Appl. Math. Lett. 2010, 23: 161–164. 10.1016/j.aml.2009.09.005MathSciNetView ArticleGoogle Scholar
- Chen C-P, Li L: Two accelerated approximations to the Euler-Mascheroni constant. Sci. Magna 2010, 6: 102–110.Google Scholar
- Chen C-P, Mortici C: New sequence converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 2012, 64: 391–398. 10.1016/j.camwa.2011.03.099MathSciNetView ArticleGoogle Scholar
- Chen C-P, Srivastava HM, Li L, Manyama S: Inequalities and monotonicity properties for the psi (or digamma) function and estimates for the Euler-Mascheroni constant. Integral Transforms Spec. Funct. 2011, 22: 681–693. 10.1080/10652469.2010.538525MathSciNetView ArticleGoogle Scholar
- DeTemple DW: A quicker convergence to Euler’s constant. Am. Math. Mon. 1993, 100: 468–470. 10.2307/2324300MathSciNetView ArticleGoogle Scholar
- Mortici C: On new sequences converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 2010, 59: 2610–2614. 10.1016/j.camwa.2010.01.029MathSciNetView ArticleGoogle Scholar
- Mortici C: Improved convergence towards generalized Euler-Mascheroni constant. Appl. Math. Comput. 2010, 215: 3443–3448. 10.1016/j.amc.2009.10.039MathSciNetView ArticleGoogle Scholar
- Negoi T: A faster convergence to the constant of Euler. Gaz. Mat., Ser. A 1997, 15: 111–113. (in Romanian)Google Scholar
- Vernescu A: A new accelerated convergence to the constant of Euler. Gaz. Mat., Ser. A 1999, 96(17):273–278. (in Romanian)Google Scholar
- Villarino M: Ramanujan’s harmonic number expansion into negative powers of a triangular number. J. Inequal. Pure Appl. Math. 2008., 9: Article ID 89. Available online at http://www.emis.de/journals/JIPAM/images/245_07_JIPAM/245_07.pdfGoogle Scholar
- Yang S: On an open problem of Chen and Mortici concerning the Euler-Mascheroni constant. J. Math. Anal. Appl. 2012, 396: 689–693. 10.1016/j.jmaa.2012.07.007MathSciNetView ArticleGoogle Scholar
- Berndt B 5. In Ramanujan’s Notebooks. Springer, New York; 1998.View ArticleGoogle Scholar
- Abramowitz M, Stegun IA (Eds): Applied Mathematics Series 55 In Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 9th edition. National Bureau of Standards, Washington; 1972.Google Scholar
- Mortici C: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 2010, 23: 97–100. 10.1016/j.aml.2009.08.012MathSciNetView ArticleGoogle Scholar
- Mortici C: Product approximations via asymptotic integration. Am. Math. Mon. 2010, 117: 434–441. 10.4169/000298910X485950MathSciNetView ArticleGoogle Scholar
- Alzer H: On some inequalities for the gamma and psi functions. Math. Comput. 1997, 66: 373–389. 10.1090/S0025-5718-97-00807-7MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.