Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma
e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma /
(x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where
$\epsilon=1$ or $\epsilon=-1$. These expansions are applied to obtain precise
inequalities for Mathieu series.

Keywords: Asymptotic expansion, residues, generalized Mathieu series,
inequalities.