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Usually the asymptotic behavior for large orders of the perturbation theoryis 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 anextremely fast convergence to the asymptotic form. A comparison of theavailable 5-loop result for this quantity with the asymptotic value showsagreement at the 0.1% level. An analogous superfast convergence to theasymptotic form happens in the case of the O(N)-symmetric anharmonic oscillatorwhere this convergence has inverse factorial type. The large orders of the$\epsilon$ expansion for critical exponents manifest a similar effect.

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