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The aim of this manuscript is to present the proof given by Michel M\'erigotin 1974 for an enlarged convergence domain of theCampbell-Baker-Hausdorff-Dynkin series in the Lie algebra of a Banach-Liegroup. This proof is based on a theorem, of independent interest, on thelifetime of the solution of a Cauchy problem. We furnish all the details forthis ODE result in Appendix A.

On M. M\'erigot's theorem on the convergence domain of the Campbell-Baker-Hausdorff-Dynkin series