Asymptotic behavior of the partial sum of the series of large deviations probabilities

Let $\{X_k\}_{k=1}^\infty$ be a sequence of independent symmetric random variables with a characteristic functions $f_k(t)$, $S_n=\sum_{k=1}^n X_k$. The asymptotic behavior of the sum $\sum_{n=1}^N\Prob\{|S_n|>n\varepsilon\}$ is investigated (for an arbitrary $\varepsilon>0$)) in the asumption that $f_k(t)$ belongs to the domain of attraction of the stable law with the index $\alpha$ ($0<\alpha\leq2$).