Limits and inequalities associated with the Euler-Mascheroni constant

作者：Chen, Chao-Ping; Mortici, Cristinel^*

来源：Applied Mathematics and Computation, 2013, 219(18): 9755-9761.

DOI：10.1016/j.amc.2013.03.089

• Euler-Mascheroni constant
• Harmonic numbers
• Inequality
• Psi function
• Polygamma functions
• Asymptotic expansion

i) We present several limits associated with the Euler-Mascheroni constant. (ii) Let gamma = 0: 577215... be the Euler-Mascheroni constant, and let T-n = Sigma(n)(k) 1/1k - ln (n+1/2+1/24n) and P-n = Sigma(n)(k-1) 2/2k-1-In (4n). We determine the best possible constants alpha,beta, a and b such that the inequalities
1/48(n+alpha)(3)<= gamma -T-n < 1/48(n+beta)(3)
and
1/24(n+a)(2)<= P-n- gamma < 1/24(n+b)(2)
are valid for all integers n >= 1.

• 出版日期2013-5-15
• 单位河南理工大学