  Non-global logarithms (NGLs) are the leading manifestation of correlations
between distinct phase space regions in QCD and gauge theories and have proven
a challenge to understand using traditional resummation techniques. Recently,
the dressed gluon expansion was introduced that enables an expansion of the NGL
series in terms of a "dressed gluon" building block, defined by an all-orders
factorization theorem. Here, we clarify the nature of the dressed gluon
expansion, and prove that it has an infinite radius of convergence as a
solution to the leading logarithmic and large-$N_c$ master equation for NGLs,
the Banfi-Marchesini-Smye (BMS) equation. The dressed gluon expansion therefore
provides an expansion of the NGL series that can be truncated at any order,
with reliable uncertainty estimates. In contrast, manifest in the results of
the fixed-order expansion of the BMS equation up to 12-loops is a breakdown of
convergence at a finite value of $\alpha_s$log. We explain this finite radius
of convergence using the dressed gluon expansion, showing how the dynamics of
the buffer region, a region of phase space near the boundary of the jet that
was identified in early studies of NGLs, leads to large contributions to the
fixed order expansion. We also use the dressed gluon expansion to discuss the
convergence of the next-to-leading NGL series, and the role of collinear
logarithms that appear at this order. Finally, we show how an understanding of
the analytic behavior obtained from the dressed gluon expansion allows us to
improve the fixed order NGL series using conformal transformations to extend
the domain of analyticity. This allows us to calculate the NGL distribution for
all values of $\alpha_s$log from the coefficients of the fixed order expansion.
