Series convergence help

Question

If I have an increasing sequence of integers $$a_n$$ and $$ \sum_{n=1}^{\infty}\frac{\ln(a_n)}{a_n}$$ diverges, does $$\sum_{n=1}^{\infty}\frac{1}{a_n}$$ also diverge

Answer 1

For a counter example, consider $$a_{n-1} = n \ln^2(n)$$ Note that $\displaystyle \sum_{n=1}^{\infty} \dfrac1{a_n}$ converges from here. Whereas $$\displaystyle \sum_{n=1}^{\infty} \dfrac{\ln(a_n)}{a_n} = \sum_{n=2}^{\infty} \dfrac{\ln(n \ln^2(n))}{n \ln^2(n)} = \sum_{n=2}^{\infty} \dfrac{\ln(n)+ \ln( \ln^2(n))}{n \ln^2(n)} = \underbrace{\sum_{n=2}^{\infty} \dfrac1{n \ln (n)}}_{\text{Diverges}} + \sum_{n=2}^{\infty} \dfrac{\ln( \ln^2(n))}{n \ln^2(n)}$$