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This paper deals with absolute convergence of real-valued rational series, i.e. mappings r:@S^*-R computed by weighted automata. An algorithm is provided, that takes a weighted automaton A as input and halts if and only if the corresponding series r"A is absolutely convergent: hence, absolute convergence of rational series is semi-decidable. A spectral radius-like parameter @r"|"r"| is introduced, which satisfies the following property: a rational series r is absolutely convergent iff @r"|"r"|