What's a good class of functions for bounding/comparing ratios of complicated logarithms? I have this goofy series $\sum \limits_{n=2}^\infty \frac{ \log_2 \left[ n \log_2^2 n \right]}{n \log_2^2 n}$ that Wolfram Alpha tells me diverges by the comparison test (and indeed, in the larger problem I'm working on I need to prove that the expression containing it diverges), but I'm struggling to find a good divergent lower bound.
I can't just throw away all the inner logarithms--then I run into the theorem $\lim \limits_{x \rightarrow \infty} \frac{\log_a x}{x^b} \rightarrow 0 \;\forall b > 0$.
So I'm looking at things like $\frac{\log_2 \left[ n^3 \right]}{n \log_2^2 n}$ (larger, unfortunately), $\frac{\log_2 \left[ n^3 \right]}{n^2 \log_2 n}$ (converges), and $\frac{\log_2 \left[ n^2 \right]}{n \log_2^2 n}$ (still too large).
Is there a more general class or form I can use to find a simple divergent lower bound, instead of stabbing in the dark?

Logarithms are really small, compared to any polynomial (or any positive power of $n$). So one very frequently useful trick is to try to ignore the logarithm, since it's not going to have a huge affect on the convergence of the series. More precisely, we have that
$$n \log_2^2 n \ge n$$
by a little bit, so let's just test the series with $n$, rather than the complicated argument inside the logarithm. This leads to 
$$\sum_n \frac{\log_2 [n \log_2^2 n]}{n \log_2^2 n} \ge \sum_n \frac{\log_2 n}{n \log_2^2 n} = \sum_{n} \frac{1}{n \log_2 n} = \infty$$

Another thing that suggests this approach is to rewrite the numerator as
$$\log_2 n + \log_2 \Big(\log_2^2 n\Big) = \log_2 n + 2 \log_2 \log_2 n$$
If $\log_2 n$ is small relative to $n$, then $\log_2 \log_2 n$ is tiny in comparison. This strongly suggests comparison to the series without this term.
