On a two-parameter class of sequences converging to a power of the base of the natural logarithm

Abstract

We investigate the following two-parameter class of real sequences (Formula presented.) where p≥0 and α>−1. In the case p≥0 and α>0, we give some sufficient and necessary conditions such that an(p)(α)≤eα for every n≥k, for each fixed k∈N, some sufficient and necessary conditions such that eα≤an(p)(α) for every n∈N, and some sufficient conditions so that the sequence strictly increasingly converges to eα, improving some results in the literature. Beside this, we give several remarks and comments related to the class of sequences and functions which are used in this investigation.