Is an effective Lagrangian a convergent series?

We present some generic arguments demonstrating that an effective Lagrangian $L_{\mathrm{eff}}$ which, by definition, contains operators $O^n$ of arbitrary dimensionality in general is not convergent, but rather an asymptotic series. It means that the behavior of the far distant terms has a specific factorial dependence $L_{\mathrm{eff}}\sim {\Sigma }_n\left(\frac{c_n{O}^n}{M^n}\right)$, $c_n\sim n!$, $n\gg 1$. We explain the main ideas by using QED as a toy model. However we expect that the obtained results have a much more general origin. We speculate on possible applications of these results to various physical problems with typical energies from 1 GeV to the Planck scale.