Showing $\sum_{n=2}^{\infty} \frac{1}{n \ln(n^3)}$ diverges

Question

Two questions:

Show whether $\sum_{n=2}^{\infty} \frac{1}{n \ln(n^3)}$ converges or diverges.

According to wolfram, the series diverges by the comparison test, so I tried the following:

for $n$ large, we get $n < n \ln(n^3) $ but then $\frac{1}{n} > \frac{1}{n \ln(n^3)}$, which gives me nothing, any tips?

Answer 1

Note that

$$\frac{1}{n \ \ln(n^3)}=\frac{1}{3n \ \ln(n)}$$

which diverges by limit comparison test with $\sum \frac{1}{n \ \ln(n)} $ which can be proved by integral test or Cauchy condensation test.