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A complete analysis is presented of the radius of convergence of the cascade connection of two SISO analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions on the coefficients of the generating series for the component systems, the radius of convergence of the cascaded system is computed. The key idea behind the analysis is to use the notion of exchangeable series in combination with the composition product.