Does the series $\sum\limits_{n=2}^\infty \frac{\ln\ln n}{\ln n}$ converge or diverge

Question

I was looking at an old Upenn calc exam and saw this question:
Does the series $\sum\limits_{n=2}^\infty \dfrac{\ln\ln n}{\ln n}$ converge or diverge? Using Wolframalpha I was able to see that it diverges using the comparison test, but I'm not sure what series to compare it to. Perhaps the harmonic series since $\ln\ln n \gt1$ and $\ln n \lt n$. Is that correct?

Answer 1

For $n > e^e$

$$\frac{\ln \ln n}{\ln n} > \frac{1}{\ln n} > \frac{1}{n}$$

since $\ln \ln n$ is strictly increasing and $\ln n < n$. The sum of a converging series (the sum of the terms before $e^e$) and a diverging series is diverging, so this series diverges.