Asymptotic behavior of the Gell-Mann-Low function in quantum field theory

The reconstruction of the Gell-Mann-Low function in quantum field theory from its asymptotic series, whose first terms are calculated using perturbation theory, is discussed. This mathematical problem cannot be solved uniquely. Nevertheless, the desired function can be reconstructed in a certain finite range of coupling constant $g$ under reasonable assumptions about this function. However, attemps to determine the behavior of the function for $g\to\infty$ are, in our opinion, groundless. Conditions under which the sum of the divergent perturbation series can rapidly decrease at infinity are determined.