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

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.