Absolute convergence of rational series is semi-decidable

This paper deals with absolute convergence of real-valued rational series, i.e. mappings r:Σ∗→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 rA is absolutely convergent: hence, absolute convergence of rational series is semi-decidable. A spectral radius-like parameter ρ|r| is introduced, which satisfies the following property: a rational series r is absolutely convergent iff ρ|r|<1. We show that if r is rational, then ρ|r| can be approximated by convergent upper estimates. Then, it is shown that the sum ∑w∈Σ∗|r(w)| can be estimated to any accuracy rate. This result can be extended to any sum of the form ∑w∈Σ∗|r(w)|p, for any integer p.