What's a good class of functions for bounding/comparing ratios of complicated logarithms?

I have this goofy series $\sum _{n=2}^{\mathrm{\infty}}\frac{{\mathrm{log}}_{2}[n{\mathrm{log}}_{2}^{2}n]}{n{\mathrm{log}}_{2}^{2}n}$ 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 $\underset{x\to \mathrm{\infty}}{lim}\frac{{\mathrm{log}}_{a}x}{{x}^{b}}\to 0\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall}b>0$

So I'm looking at things like $\frac{{\mathrm{log}}_{2}\left[{n}^{3}\right]}{n{\mathrm{log}}_{2}^{2}n}$ (larger, unfortunately), $\frac{{\mathrm{log}}_{2}\left[{n}^{3}\right]}{{n}^{2}{\mathrm{log}}_{2}n}$ (converges), and $\frac{{\mathrm{log}}_{2}\left[{n}^{2}\right]}{n{\mathrm{log}}_{2}^{2}n}$ (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?

I have this goofy series $\sum _{n=2}^{\mathrm{\infty}}\frac{{\mathrm{log}}_{2}[n{\mathrm{log}}_{2}^{2}n]}{n{\mathrm{log}}_{2}^{2}n}$ 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 $\underset{x\to \mathrm{\infty}}{lim}\frac{{\mathrm{log}}_{a}x}{{x}^{b}}\to 0\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall}b>0$

So I'm looking at things like $\frac{{\mathrm{log}}_{2}\left[{n}^{3}\right]}{n{\mathrm{log}}_{2}^{2}n}$ (larger, unfortunately), $\frac{{\mathrm{log}}_{2}\left[{n}^{3}\right]}{{n}^{2}{\mathrm{log}}_{2}n}$ (converges), and $\frac{{\mathrm{log}}_{2}\left[{n}^{2}\right]}{n{\mathrm{log}}_{2}^{2}n}$ (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?