Asymptotic of expansion of the evolution operator kernel in powers of time interval $\Delta t$

The upper bound for asymptotic behavior of the coefficients of expansion of the evolution operator kernel in powers of the time interval $\Delta t$ was obtained. It is found that for the nonpolynomial potentials the coefficients may increase as $n!$. But increasing may be more slow if the contributions with opposite signs cancel each other. Particularly, it is represented an example of the potential, for which the expansion converges. For the polynomial potentials $\Delta t$-expansion is certainly asymptotic one. The coefficients increase in this case as $\Gamma (n \frac {L-2}{L+2})$, where $L$ is the order of the polynom.