Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality

摘要

Based on the Padé approximation method, in this paper we determine the coefficients aj and bj (1 ≤ j ≤ k) such that 1e(1+1x)x=xk+a1xk−1+⋯+akxk+b1xk−1+⋯+bk+O(1x2k+1),x→∞, where k ≥ 1 is any given integer. Based on the obtained result, we establish new upper bounds for (1+1/x)^x. As an application, we give a generalized Carleman-type inequality.