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We prove a convergence result for the Campbell–Baker–Hausdorff–Dynkin seriesin infinite-dimensional Banach–Lie algebras. In the existing literature, this topic has been investigated whenis the Lie algebra of a finite-dimensional Lie group(see [Blanes and Casas, 2004]) or of an infinite-dimensional Banach–Lie group(see [Mérigot, 1974]). Indeed, one can obtain a suitable ODE for, which follows from the well-behaved formulas for the differential of the Exponential Map of the Lie group. The novelty of our approach is to derive this ODE in any infinite-dimensional Banach–Lie algebra, not necessarily associated to a Lie group, as a consequence of an analogous abstract ODE first obtained in the most natural algebraic setting: that of the formal power series in two commuting indeterminatesover the free unital associative algebra generated by two non-commuting indeterminates. [ABSTRACT FROM AUTHOR]