Use the Limit Comparison Test or Comparison Test Instead of Integral Test

Question

For the infinite series $$\sum_{n=4}^{\infty} \frac{1}{n\ln(n^2)} $$
I found a solution with the integral test that proves its divergence. However, I was expected to use either the limit comparison test or comparison test instead of the integral test. I can't quite figure out how to apply either test in this situation.

Answer 1

Have a look at Bertrand series http://www.sosmath.com/calculus/series/bertrand/bertrand.html