Prove that $\sum\frac{(\log n)^2}{n^3}$ converges This question is from Serge Lang's textbook, in a chapter that comes before the ratio and integral tests are introduced, so those can't be used. I've already proved that $\sum\frac{\log n}{n^3}$ converges and have an inkling that this result may be useful, but I can't figure out how.

$$
\forall \varepsilon>0, \,\, \sum_{n\ge 1}\frac{1}{n^3}<\sum_{n\ge 1}\frac{(\ln n)^2}{n^3} \le \sum_{n\ge 1}\frac{1}{n^{3-\varepsilon}}+O(1).
$$
