While trying to prove some series (which should be easy to guess) were divergent, I stumbled across the following cute little result. We write $\log^{(k)}x = \log \log \log .... \log x$ with $k \log$s.

Consider the integral $\int_a^b \frac{dt}{t \log t \log \log t ... \log^{(n)}t}$.

Then the substitution $t=e^s$ gives this integral as equal to $\int_{\log a}^{\log b} \frac{ds}{s \log s \log \log s .... \log^{(n-1)} s}$, so making $n$ such substitutions will just reduce this integral to solving $\int \frac{dx}{x}$, which is well-known.