In (Cacciagrano, et al., 2008) the authors studied the expressiveness of persistence in the asynchronous π-calculus, henceforth Aπ. They considered Aπ and three sub-languages of it, each capturing one source of persistence: the persistent-input calculus (PIAπ), the persistent-output calculus (POAπ), and the persistent calculus (PAπ). They prove that, under some general conditions, there cannot be an encoding from Aπ into a (semi)-persistent calculus preserving the must-testing semantics, a semantics sensitive to divergence. In this paper we support and strengthen the separation results of (Cacciagrano, et al., 2008) by showing that convergence and divergence are two decidable properties in a fragment of POAπ and PAπ, in contrast to what happen in Aπ. Thus, it is shown that there cannot be a (computable) encoding from Aπ into PAπ and in such a fragment of POAπ, preserving divergence or convergence. These impossibility results don’t presuppose any condition on the encodings and involve directly convergence for first time in the study of the expressiveness of persistence of .