  We investigate some topics related to the celebrated Baker-Campbell-Hausdorff
Theorem: a non-convergence result and prolongation issues. Given a Banach
algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we
study the relationship of different issues: the convergence of the BCH series
$\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the
convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$
which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by
suitable matrix counterexamples, we show that various pathologies can occur,
among which we provide a non-convergence result for the BCH series. This
problem is related to some recent results, of interest in physics, on closed
formulas for the BCH series: while the sum of the BCH series presents several
non-convergence issues, these closed formulas can provide a prolongation for
the BCH series when it is not convergent. On the other hand, we show by
suitable counterexamples that an analytic prolongation of the BCH series can be
singular even if the BCH series itself is convergent.
