Superfast convergence effect in large orders of the perturbative and $epsilon$ expansions for the O(N) symmetric $phi^4$ model

Usually the asymptotic behavior for large orders of the perturbation theory is reached rather slowly. However, in the case of the N-component $phi^4$ model in D=4 dimensions one can find a special quantity that exhibits an extremely fast convergence to the asymptotic form. A comparison of the available 5-loop result for this quantity with the asymptotic value shows agreement at the 0.1% level. An analogous superfast convergence to the asymptotic form happens in the case of the O(N)-symmetric anharmonic oscillator where this convergence has inverse factorial type. The large orders of the $epsilon$ expansion for critical exponents manifest a similar effect.