I have a convergent series of positive terms $\sum_{n=1}^{\infty} a_n$ and have to check if series $$ \sum_{n=1}^{\infty}\frac{\sqrt{a_n}}{\log(n)}\left(n^{a_n}-1\right)$$ converges. I think it does, but don't know how to prove that, I tried using Cauchy criterion, but couldn't really make it work.
Convergence of series depending on convergent series
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sequences-and-series
convergence