A double inequality for a generalized-Euler-constant function

作者：Lampret Vito^*

来源：Journal of Mathematical Analysis and Applications, 2011, 381(1): 155-165.

DOI：10.1016/j.jmaa.2011.03.049

• Acceleration of convergence
• Generalized Euler's constants
• Generalized gamma-sequence

The rate of convergence of the sequence n bar right arrow gamma(n)(a) := Sigma(k=0) (n-1) 1/a+k - In a+n-1/a, a > 0, towards the generalized Euler's constant gamma(a) := lim(n ->infinity) gamma(n)(a), where gamma(1) is the Euler-Mascheroni constant, is accurately estimated using the Euler-Maclaurin summation formula. The expression
gamma(a) = S(n)* (a, q) + R(n)* (a, q)
with parameters n, q is an element of N. where S(n)*(a, q) is a sum consisting of n + 3q + 2 summands and R(n)*(a, q) is a remainder, is derived. The error term is estimated as
vertical bar B2q vertical bar /2(a+n+2)2q+1(-1)q-1R(n)*(a, q) < (1 - 4-q) vertical bar B2q vertical bar/(a + n)2q+1
12 root 9/a+n (q/e pi(a+n))(2q),
where B(k) is the kth Bernoulli coefficient. Two similar expressions are also established.

• 出版日期2011-9-1