We study the analytic properties of the t Hooft coupling expansion of the beta-function at the leading nontrivial large-$N_f$ order for QED, QCD, Super QED and Super QCD. For each theory, the t Hooft coupling expansion is convergent. We discover that an analysis of the expansion coefficients to roughly 30 orders is required to establish the radius of convergence accurately, and to characterize the (logarithmic) nature of the first singularity. We study summations of the beta-function expansion at order $1/N_f$, and identify the physical origin of the singularities in terms of iterated bubble diagrams. We find a common analytic structure across these theories, with important technical differences between supersymmetric and non-supersymmetric theories. We also discuss the expected structure at higher orders in the $1/N_f$ expansion, which will be in the future accessible with the methods presented in this work, meaning without the need for resumming the perturbative series. Understanding the structure of the large-$N_f$ expansion is an essential step towards determining the ultraviolet fate of asymptotically non-free gauge theories.
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