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.

Well then use $\ln(n)^2 = 4\ln(\sqrt{n})^2 < 4\sqrt{n}^2$.