Solution to the Problem of Monotonicity of a Class of Sequences Converging to lnp

Abstract

We completely describe the monotonicity of the following class of real sequences, $z_n^{\left(\alpha \right)}:=(1-\alpha ){\sum }_{j=n+1}^{pn}\frac{1}{j}+\alpha {\sum }_{j=n}^{pn}\frac{1}{j},n\in \mathbb{N}$, where $p$ is a natural number bigger or equal to two and the parameter $\alpha$ belongs to the interval $[0,1]$, extending and unifying many results in the literature in an elegant way. Here, the monotonicity refers to the monotonicity character of each sequence belonging to the class on the whole domain of indices (i.e., on the set $\mathbb{N}$), not only to the eventual monotonicity of the sequences, which was the case in many previous investigations.