科研之友
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摘要
We introduce a new class of sequences of the form
mu(n) = Sigma(n)(k=1)1/k + ln(e(a/n+b)) - 1) - ln a
which converge to the Euler-Mascheroni constant gamma. Being preoccupied to accelerate the classical sequence convergent toward gamma, Batir [J. Ineq. Pure Appl. Math. 6 (2005) no. 4 Art 1031 and Alzer [Expo. Math. 24 (2006) 385-388] studied the case a = b = 1 and we show in this paper that the fastest sequence (mu(n))(n >= 1) is obtained for a = 1/root 2, b = (2 + root 2)/4. For these values, accurate approximations of gamma can be constructed, as numerical computations made in the final part of this paper show. We also solve an open problem about the rate of convergence of some sequences defined by Batir.
- 出版日期2010
