Accurate estimates of (1+x)^{1/x} Involved in Carleman Inequality and Keller Limit

TL;DR

This paper generalizes inequalities involving (1+x)^{1/x} using Maclaurin series, introduces new Keller-type limits, and extends known results in the context of Carleman inequalities.

Contribution

It provides a novel generalization of inequalities and limits related to (1+x)^{1/x} using Maclaurin series, expanding the theoretical framework.

Findings

Generalized inequalities from previous works.

Defined new Keller-type limits.

Extended known results in the field.

Abstract

In this paper, using the Maclaurin series of the functions $(1+x)^{1/x}$, some inequalities from papers Bicheng Debnath [1998] and Mortici Jang [2015] are generalized. For arbitrary Maclaurin series some general limits of Keller's type are defined and applying for generalization of some well known results.