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.

Hint: Prove $(\log n)^2 <  n$ for large enough $n$ and compare your series to $\sum\frac{1}{n^2}$.
