The rate of convergence for an almost surely convergent series $S_n={Sigma^n}_{i-1}X_i$ of independent random variables is studied in this paper. More specifically, when S$_{n}$ converges almost surely to a random variable S, the tail series $T_n{equiv}$ S - S_{n-1} = {Sigma^infty}_{i-n} X_i$</TEX> is a well-defined sequence of random variables with T$_{n}$ $ ightarrow$ 0 almost surely. Conditions are provided so that for a given positive sequence {$b_n, n {geq$ 1}, the limit law sup$_{kappa}geqn | T_{kappa}|/b_n ightarrow$ 0 holds. This result generalizes a result of Nam and Rosalsky [4].