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

We investigate the following two-parameter class of real sequences $$ a_{n}^{(p)}(\alpha )=\Big(1+\frac{\alpha }{n}\Big)^{n+p},\quad n\in { \mathbb{N}}, $$ where $p\ge 0$ and $\alpha >-1$ . In the case $p\ge 0$ and $\alpha >0$ , we give some sufficient and necessary conditions such that $a_{n}^{(p)}(\alpha )\le e^{\alpha }$ for every $n\ge k$ , for each fixed $k\in {\mathbb{N}}$ , some sufficient and necessary conditions such that $e^{\alpha }\le a_{n}^{(p)}(\alpha )$ for every $n\in {\mathbb{N}}$ , and some sufficient conditions so that the sequence strictly increasingly converges to $e^{\alpha }$ , 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.