Precise asymptotics over a small parameter for a series of large deviation probabilities

We obtain the asymptotics of the series
$$ \sum^\infty_{k=1}w_k({\mathbf P}(|S_k|\geq\varepsilon_k) $$
are par as $\varepsilon\downarrow0,$ where $S_k$ tial sums of independent and identically distributed random variables in the domain of attraction of a non-degenerate stable law, $w$ and $\varepsilon$ are regularly varying functions (in Karamata’s sense).