On the convergence rate in precise asymptotics

We study some aspects of estimation of the convergence rate in the so-called “exact asymptotics.” In particular, we obtain asymptotic expansions in powers of $\varepsilon$ of sums of the form $\sum_{n\ge 1} n^s\,\mathbf P(\xi_{\alpha}> \varepsilon n^{\delta})$, where a random variable $\xi_{\alpha}$ has a stable distribution with an exponent $\alpha\in (0, 2]$, $\delta>0$, $s\in \mathbf R$.