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 a j $a_{j}$ and b j $b_{j}$ ( 1 ≤ j ≤ k $1leq j leq k$ ) such that 1 e ( 1 + 1 x ) x = x k + a 1 x k − 1 + ⋯ + a k x k + b 1 x k − 1 + ⋯ + b k + O ( 1 x 2 k + 1 ) , x → ∞ , $$ rac{1}{e} iggl( 1+rac{1}{x} iggr) ^{x}= rac{x^{k}+a_{1}x^{k-1}+ cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+cdots +b_{k}}+O iggl( rac{1}{x ^{2k+1}} iggr) , quad x