Asymptotic expansions of several series and their application

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Asymptotic expansions of series ∑ₖ₌₀∞ εᵏ(k+a)γe⁻⁽ᵏ⁺ᵃ⁾αˣ and ∑ₖ₌₀∞ εᵏ(k+a)γ/ (x(k+a)α+1)μ in powers of x as x→+0 are found, where ε=1 or ε=-1. These expansions are applied to obtain precise inequalities for Mathieu series. Keywords: Asymptotic expansion, residues, generalized Mathieu series, inequalities.