It is known that $\sum_{n=1}^\infty \frac{1}{n}=\infty$, $\sum_{n=1}^\infty \frac{1}{n\ln(n)}=\infty$, $\sum_{n=1}^\infty \frac{1}{n\ln(n)\ln(\ln(n))}=\infty$ etc.
But what happens if we consider the following series?
$$\sum_{n=1}^\infty \frac{1}{n\prod_{k=1}^{a(n)}ln^k(n)}$$
(with $ln^k(x)$ describing k iterations and $a(n)$ the maximum possible value such that $ln^{a(n)}(n)>1$
If this is convergent, what function desribes its rate of convergence (i.e. minimum number of terms to reach a given precision), and if it is divergent, what is its rate of divergence (i.e. minimum number of terms to pass a natural number $n$)?
Edit to fix André Nicolas' justified objection.
Edit the second: This series diverges... see Putnam 2008 A4. That does leave the question about rate of divergence, though!