studying the series $\sum_\limits{n=1}^\infty \frac{1}{n^ {\alpha}(\log n)^ {\beta}}$.
On convergence of Bertrand series $\sum\limits_{n=2}^{\infty} \frac{1}{n^{\alpha}\ln^{\beta}(n)}$ where $\alpha, \beta \in \mathbb{R}$