We review and provide simplified proofs related to the Magnus expansion, and
improve convergence estimates. Observations and improvements concerning the
Baker--Campbell--Hausdorff expansion are also made. In this Part I, we consider
the general Banach algebraic setting. We show that the (cumulative) convergence
radius of the Magnus expansion is 2; and of the Baker--Campbell--Hausdorff
series is C2​=2.89847930….Comment: Part I of original submission arXiv:1709.01791v1, rewritten and
expande