Logarithmic convergence criterion

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

A criterion for the convergence of series $\sum a_n$ of positive real numbers. If there are $\alpha > 0$ and $N$ such that \begin{equation}\label{e:compare1} \frac{\ln (a_n)^{-1}}{\ln n} \geq 1 + \alpha \qquad \forall n\geq N \end{equation} then the series converges. If there is $N$ such that \begin{equation}\label{e:compare2} \frac{\ln (a_n)^{-1}}{\ln n} \leq 1 \qquad \forall n \geq N \end{equation} then the series diverges. Indeed \eqref{e:compare1} implies that $\sum_n a_n$ is dominated by $\sum \frac{1}{n^{1+\alpha}}$ (namely that \[ \left.a_n \leq \frac{1}{n^{1+\alpha}}\right)\, , \] whereas \eqref{e:compare2} implies that $\sum_n a_n$ dominates the harmonic series.