A double inequality for a generalized-Euler-constant function

Abstract

The rate of convergence of the sequence n ↦ γ n ( a ) : = ∑ k = 0 n − 1 1 a + k − ln a + n − 1 a , a > 0 , towards the generalized Eulerʼs constant γ ( a ) : = lim n → ∞ γ n ( a ) , where γ ( 1 ) is the Euler–Mascheroni constant, is accurately estimated using the Euler–Maclaurin summation formula. The expression γ ( a ) = S n ⁎ ( a , q ) + R n ⁎ ( a , q ) with parameters n , q ∈ N , where S n ⁎ ( a , q ) is a sum consisting of n + 3 q + 2 summands and R n ⁎ ( a , q ) is a remainder, is derived. The error term is estimated as | B 2 q | 2 ( a + n + 2 ) 2 q + 1 < ( − 1 ) q − 1 R n ⁎ ( a , q ) < ( 1 − 4 − q ) | B 2 q | ( a + n ) 2 q + 1 < 12 q a + n ( q e π ( a + n ) ) 2 q , where B k is the kth Bernoulli coefficient. Two similar expressions are also established.