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Abstract: This paper deals with absolute convergence of real-valued rational series, i.e. mappings computed by weighted automata. An algorithm is provided, that takes a weighted automaton as input and halts if and only if the corresponding series is absolutely convergent: hence, absolute convergence of rational series is semi-decidable. A spectral radius-like parameter is introduced, which satisfies the following property: a rational series is absolutely convergent iff . We show that if is rational, then can be approximated by convergent upper estimates. Then, it is shown that the sum can be estimated to any accuracy rate. This result can be extended to any sum of the form , for any integer . [Copyright &y& Elsevier]