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

Abstract

Based on the Padé approximation method, in this paper we determine the coefficients and () such that where is any given integer. Based on the obtained result, we establish new upper bounds for . As an application, we give a generalized Carleman-type inequality.

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

“…The above result immediately shows that (−1) j a j > 0 so that (3.7) is an alternating series for positive x. Recently, Chen and Paris [10] obtained a recurrence relation for β j = (−1) j a j given by…”

“…Chen and Paris [10] have given an integral representation for the coefficients β j and have proved that the sequence {β j } ∞ j=0 is monotonically decreasing. They thereby obtained the following double inequality [10, Theorem 2.1]:…”