Let ∑n=1∞Xn be a series of independent random variables with at least one non-degenerate Xn, and let Fn be the distribution function of its partial sums Sn=∑k=1nXk. Motivated by Hildebrand's work in [1], the authors investigate the a.s. convergence of ∑n=1∞Xn under a hypothesis that ∑n=1∞ρ(Xn,cn)=∞ whenever ∑n=1∞cn diverges, where the notation ρ(X, c) denotes the Lévy distance between the random variable X and the constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of Fn(x0) with the limit value 0 <L0 < 1, together with the essential convergence of ∑n=1∞Xn, is both sufficient and necessary in order for the series ∑n=1∞Xn to a.s. converge. Moreover, if the essential convergence of ∑n=1∞Xn is strengthened to n→∞limsupP(|Sn<K|)=1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of ∑n=1∞Xn. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand[1] as well.