Understanding a statement about the series $S =\sum_2 ^\infty \frac{1}{n\ln n}$
Solution 1:
By $n\ln n < n^{1+\epsilon}$, thus $\sum \dfrac{1}{n\ln n} > \sum \dfrac{1}{n^{1+\epsilon}}$. The series $\sum \dfrac{1}{n^{1+\epsilon}}$ converges. This cannot be conclusive about the convergence of $S$, because $S$ is bigger than something finite then we don't know if $S$ is finite or infinite. Hence, you should use integral.