Large-Order Estimates in Perturbative QCD and Non-Borel Summable Series

Basic facts about the summation of divergent power series are reviewed, both for series with non vanishing and for series with vanishing convergence radius. Particular attention is paid to the recent development that makes it possible, in the former case, to define summation in the whole Mittag-Leffler star and, in the latter case, to define summation when the point of expansion lies at the tip of a horn-shaped analyticity domain with zero opening angle. Relevance of these results to perturbative QCD is stressed in relation to current discussions concerning large-order estimates of perturbative QCD expansion coefficients.