  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 $\mathrm C_2=2.89847930\ldots$.
