Approximation formulas for the constant e and an improvement to a Carleman-type inequality

We give an explicit formula for the determination of the coefficients c(j) appearing in the expansion x(1 + Sigma(q)(j=1) c(j)/x(j)) (root pi/Gamma(x + 1/2))(1/x) = e + O (1/x(q+1)) for x -> infinity and q is an element of N: = {1, 2, ...}. We also derive a pair of recurrence relations for the determination of the constants lambda(l) and mu(l) in the expansion (1 + 1/x)(x) similar to e (1 + Sigma(infinity)(l=1) lambda l/(x + mu(l))(2l-1)) as x -> infinity. Based on this expansion, we establish an inequality for (1 + 1/x)(x). As an application, we give an improvement to a Carleman-type inequality. (C) 2018 Elsevier Inc. All rights reserved.