Absolute Convergence of Rational Series is semi-decidable
This paper deals with absolute convergence of real-valued rational series, i.e. that can be 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 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 Σ |r(w)| can be estimated to any accuracy rate. This result can be extended to any sum of the form Σ |r(w)|p, for any integer p.