Solutions to Two Problems on a Class of Sequences Converging to the Number e

Abstract

We solve two problems concerning the family of real sequences (Formula presented.), where (Formula presented.), considerably improving and complementing the results in the literature. Namely, we find all the values of the parameter (Formula presented.) such that (Formula presented.), for every (Formula presented.), and find all the values of the parameter such that the sequence (Formula presented.) is strictly increasing on the whole domain (i.e., on the set (Formula presented.)). An interesting and highly nontrivial auxiliary result about the behavior of a real function is obtained.