## AbstractWe study \emph{real-valued absolutely convergent rational series}, i.e. functions $r: \Sigma^* \rightarrow {\mathbb R}$, defined over a free monoid $\Sigma^*$, that can be computed by a multiplicity automaton $A$ and such that $\sum_{w\in \Sigma^*}|r(w)|<\infty$. We prove that any absolutely convergent rational series $r$ can be computed by a multiplicity automaton $A$ which has the property that $r_{|A|}$ is simply convergent, where $r_{|A|}$ is the series computed by the automaton $|A|$ derived from $A$ by taking the absolute values of all its parameters. Then, we prove that the set ${\cal A}^{rat}(\Sigma)$ composed of all absolutely convergent rational series is semi-decidable and we show that the sum $\sum_{w\in \Sigma^*}|r(w)|$ can be estimated to any accuracy rate for any $r\in {\cal A}^{rat}(\Sigma)$. We also introduce a spectral radius-like parameter $\rho_{|r|}$ which satisfies the following property: $r$ is absolutely convergent iff $\rho_{|r|}<1$.
[Edit] |