While messing around with some convergent series for a bit, I thought of the following problem: Let $f_n(x)$ be defined recursively by $f_0(x) = x$ and $f_n(x)= \ln(\vert f_{n-1}(x)\vert)$ and let $(a_k)$ be such that $f_n(a_k)$ is well-defined for all $k$.
Assuming $a_k\in\mathbb{R}^+$, $\sum_{k=1}^\infty {a_k}$ converges, need $\sum_{k=1}^\infty {a_k f_n(a_k)}$ converge also?
After some experimentation, I found a set of counterexample for the case $n=1$, given by $a_k = \frac{1}{k\ln(k)^{1+\epsilon}}$, where $\epsilon\in\left(0,1\right)$.
However, despite quite a bit of effort, I couldn't construct any counterexamples for $n\geq 2$. I strongly suspect that a counterexample exists for each $k$, but was unable to prove this result.