Latter research on Euler-Mascheroni constant Valentin Gabriel Cristea and Cristinel Mortici arxiv:3.4397v [math.ca] 6 Dec 03 Ph. D. Student, University Politehnica of Bucharest, Splaiul Independenţei 33, Bucharest, Romania, valentingabrielc@yahoo.com Prof. dr. habil., Valahia University of Târgovişte, Bd. Unirii 8, 3008 Târgovişte, Romania, cristinelmortici@yahoo.com Abstract: In this work, we present a review and an example on some latter results on the problem of approximating the Euler- Mascheroni constant. We use the method firstly introduced in [C. Mortici, Product Approximations via Asymptotic Integration Amer. Math. Monthly 7 5) 00) 434-44]. Keywords: sequences; Euler-Mascheroni constant; harmonic sum; Cesaro- Stolz lemma; rate of convergence MSC: 00: 6D5, Y5, 4A5, 34E05. Introduction and motivation The famous Euler-Mascheroni constant γ = 0, 5775664905386... was firstly studied by the Swiss mathematician Leonhard Euler 707-783) and the Italian mathematician Lorenzo Mascheroni 750-800) as the it of the sequence γ n = + + 3 + + n lnn. The question on γ whether it is a rational number or not, has not an answeryet. The reason seems to be the inexistence of very fast convergences to γ, having a simple form. As a consequence, in the recent past, many authors introduced new fast convergences to γ. We start a history of such approximations with the results of Tims and Tyrrell [0] Young [3] n+) < γ n γ < n ), n+) < γ n γ < n,
Anderson, Barnard, Richards, Vamanamurthy and Vuorinen [3] Mortici and Vernescu [3] γ n < γ n γ < n, or Toth [] n+ < γ n γ < n, n+ 5 < γ n γ < n+. 3 Independently, Alzer [] and Chen-Qi [4] proved n+ γ γ < γ n γ < n+. 3 Next, Qiu and Vuorinen [9, Cor..3] showed n α n < γ n γ < n β n, where α = / and β = γ /, with its consequence n 8n < γ n γ < n. This is also called Franel s inequality [8, Ex. 8]. Karatsuba [9] proved n n + 0n 4 6n 6 < γ n γ < n n + 0n 4, while Mortici [4] gave n+ 3 + 8n < γ n γ < n+ 3 +. 3n DeTemple [7, 8] introduced the sequence R n = + + 3 + + n ln n+ ) that converges to γ like n, since 4n+) < R n γ < 4n. Chen [5] found 4n+a) < R n γ < 4n+b),
with a = 4 γ+ ln 3) and b =. Chen and Mortici [6] improved these bounds to 7 4n+ 960 ) ) n+ 4 + < R n γ < 7 4n+ 960 ) ) n+ 4 + 3 8064 n+ ) 6 3 8064 ) 6 n+ For further reading, please see [3], [4], [5], [8], []. Mortici [5], introduced the sequence 7 3070 ) 8 n+ µ n a,b) = + + 3 + + n + an lnn+b), depending on real parameters a, b. This family extends the sequence V n = µ n,0) introduced by Vernescu [] and DeTemple sequence R n = µ n, ). See [7]. Both sequences Vn and R n converges as n. For proofs and details, see [7], [8], []. The results in [5] show that for a = 6 6, b = / 6 u n = + + 3 + + n + n+ 6 6)n ln 6 ) and a = 6+ 6, b = / 6 u n = + + 3 + + n + n 6+ 6)n ln 6 ) converge to γ with the speed of convergence at n 3.. A convergence towards γ Using the idea from [5], we introduce the family of sequences v n = v n a,b) v n a,b) = + + + n + an+b nn ) lnn, depending on real parameters a and b. In order to avoid some inconvenience, we assume v 0, v, v given. This is an extension of the classical convergence γ n ) n, since γ n = v n, ). Known fact, the new introduced sequence converges to γ as n in case a =, b =. 3
The problem we rise here is what are the best parameters a and b which provide the fastest sequence v n a,b). The answer is formulated as the following Theorem.. i) If a 3, then the sequence v na,b)) n has the rate of convergence n. ii) If a = 3 and b 5 then the sequence v 3 n,b)) has the rate of n convergence n. iii) If a = 3 and b = 5 then the sequence v 3 n, )) 5 has the rate n of convergence n 3. We use the following Lemma.. If the sequence x n ) n is convergent to x and there exists the it n nk x n x n+ ) = l R with k >, then there exists the it n nk x n x) = l k. This is a form of Cesaro-Stolz lemma, which is useful in constructing of asymptotic expansions, or evaluating the speed of convergence. For proof and other details, see e.g. []. Proof of Theorem.. We have v n v n+ = an+b nn ) n an+)+b n ln nn+) n+. By using a computer software such as Maple, we get v n v n+ = a 3 ) n + i) If a 3, then while Lemma. says + a 5 ) 4 n 4 + a+b 4 5 a+b 3 ) n 3 ) n 5 +O n 6 n n v n v n+ ) = a 3 0, n nv n γ) = a 3 0. As a consequence, v n ) n converges as n. ). ) 4
ii) and iii). If a = 3, then ) reads as v n v n+ = b+ 5 ) 6 n 3 + 4n 4 + b+ 7 ) ) 0 n 5 +O n 6. If b 5, then and by Lemma., n n3 v n v n+ ) = b+ 5 6 0, n n v n γ) = b+ 5 0. As a consequence, v 3 n,b)), with b 5 n, converges as n. Finally, with a = 3 and b = 5, we get from ) v n v n+ = 4n 4 ) 5n 5 +O n 6, then use Lemma. to obtain 3 n n3 v n, 5 ) ) γ =. ) Now the sequence v 3 n, )) 5 n has the rate of convergence n 3 and the theorem is proved. 3. Final remarks In fact, we obtained the sequence s n = + + + n + 3 n ) + 5 n lnn converging, according to ), as n 3, that is the fastest possible through all sequences v n a,b)) n. In this case, the best constants a = 3, b = 5 obtained in the previous sections can be obtained using another method. First remark that v n a,b) = γ n + an+b nn ) n n. Using the representation of the harmonic sum h n in terms of digamma function h n = γ + n +ψn), e.g. [, p. 58, Rel. 6.3.] and the asymptotic formula [, p. 59, Rel. 6.3.8] ψz) = lnz z z + 0z 4 5z 6 +, 5
we get γ n = h n lnn = γ + n n n + 0n 4 5n 6 +. Thus v n a,b) = γ + a 3 )n + ) b+ 5 n+ n n ) + 0n 4 5n 6 +. By analyzing the first fraction in the above representation, the fastest sequence v n a,b)) n isobtainedwhenthecoefficientsa 3 andb+ 5 vanishsimultaneously. On the other hand, let us note that ) offers us the approximation We prove the following s n γ n3, as n. and Theorem.. For every integer n 9, we have n 3 + 0n 4 < s n γ < n 3 + 3 0n 4. The left hand side inequality holds for every integer n 3. Proof. The sequences z n = s n γ) n 3 + ) 0n 4 t n = s n γ) n 3 + 3 ) 0n 4 converges to zero. In order to prove z n > 0 and t n < 0, it suffices to show that z n ) n 3 is decreasing and t n ) n 9 is increasing. As s n+ s n = 3n n ) + 5 n+) ln + ), n we get z n+ z n = f n) and t n+ t n = gn), where f x) = 3x x ) + 5 ) x+) 3 + 0x+) 4 x+) ln gx) = 3x x ) + 5 x+) ln 6 + ) x + x 3 + ) 0x 4 + x )
) x+) 3 + 3 0x+) 4 Using again Maple software, we obtain f x) = P x) 60x 5 x ) x+) 5 + x 3 + 3 ) 0x 4. and with g Qx) x) = 60x 5 x ) x+) 5, P x) = 60+00x )+348x ) +055x ) 3 +875x ) 4 +50x ) 5 and Qx) = 77064+75456x 9)+80376x 9) +64805x 9) 3 +7405x 9) 4 +930x 9) 5 +0x 9) 6. Evidently, f > 0 on, ) and g < 0 on 9, ). It follows that f is strictly increasing on, ) and g is strictly decreasing on 9, ). As f ) = g ) = 0, we get f < 0 on, ) and g > 0 on 9, ). It follows that z n ) n is strictly decreasing, t n ) n 9 is strictly increasing. As we explained, the conclusion follows. Acknowledgements. The work of the second author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS- UEFISCDI project number PN-II-ID-PCE-0-3-0087. References [] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 07. [] H. Alzer, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg 68 998), 363-37. [3] G. D. Anderson, R. W. Barnard, M. K. Vamanamurthy, M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 995), 73-73. [4] C. P. Chen, F. Qi, The best harmonic sequence, arxiv:math/030633, available online at: http://arxiv.org/abs/math/030633. [5] Ch.-P. Chen, Inequalities for the Euler-Mascheroni constant, Appl. Math. Lett. 3 00), 6-64. 7
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