For which values $\alpha, \ \beta \in \mathbb R $ does the series $\sum_{n=2}^{\infty} \frac{1}{n^\alpha \ln^{\beta}(n)}$ converges?
For which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}\ln^{\beta}x}$ converge?