Is this series $\sum_{n \geq 2}\sqrt{a_n}\frac{n^{a_n}-1}{\ln n}$ always divergent? Let $\displaystyle \sum a_n$ be a series with positive terms which is a convergent series and suppose that we don't have $\displaystyle a_n \ln n \rightarrow 0$.
Is the following series always divergent?
$$ B=\displaystyle  \sum_{n \geq 2} \sqrt{a_n}  \:\frac{n^{a_n}-1}{\ln{n}}$$
I've tried with no success so far to find some series $\displaystyle \sum a_n$ with the above conditions for which $B$ is convergent.
An answer to this question would be an interesting extension to the answer that, with the aid of Kelenner, I've given here.

The following appears to be a counterexample:
$a_n = \dfrac{1}{\ln(n)}$ if $n = 2^{k^2}$ for $k\in\mathbb N$, and $a_n=\dfrac{1}{2^n}$ otherwise.
