Asymptotic series in quantum-field asymptotics

After a brief history of investigations into the determination of asymptotic behaviors in quantum field theory, the present state of the art is reviewed. The part played by asymptotic series in renormalization-group analysis of asymptotic behaviors is discussed, and a detailed analysis is made of the problem of reconstructing the function $\beta(g)$ from the first terms of the expansion in a power series in $g$ known from perturbation theory and the limiting expression for the coefficients of the asymptotic series at large $k$, where $k$ is the number of a term of the series, which can be obtained by the functional saddle point method. It is shown that information about the behavior at large $g$ is needed for reliable reconstruction of the sum.