State without proof the Integral Comparison Test for the convergence of a series $\sum_{n=1}^{\infty} a_{n}$ of non-negative terms.

Determine for which positive real numbers $\alpha$ the series $\sum_{n=1}^{\infty} n^{-\alpha}$ converges.

In each of the following cases determine whether the series is convergent or divergent: (i) $\sum_{n=3}^{\infty} \frac{1}{n \log n}$, (ii) $\sum_{n=3}^{\infty} \frac{1}{(n \log n)(\log \log n)^{2}}$, (iii) $\sum_{n=3}^{\infty} \frac{1}{n^{(1+1 / n) \log n}}$.