Summary: The behavior of tail probabilities \({\mathbf P}\{S\leq r\}\), \(r\to 0\), is investigated, where \(S\) is a series \(S=\sum \lambda_j Z_j\) generated by some sequence of positive numbers \(\{\lambda_j\}\) and by a sequence \(\{Z_j\}\) of independent copies of a positive random variable \(Z\). We present the exact asymptotic expression for \({\mathbf P}\{S\leq r\}\) by means of Laplace transform \(\Lambda(\gamma)={\mathbf E}\exp\{-\gamma S\}\) 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.