The behavior of tail probabilities P{S ≤ r}, r → 0 is investigated, where S is a series S = ∑λ_jZ_j generated by some sequence of positive numbers {λ_j} and by a sequence {Z_j} of independent copies of a positive random variable Z. We present the exact asymptotic expression for P{S ≤ r} by means of Laplace transform Λ(γ) = Eexp{-γS} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.