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

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