Convergence of High-Intensity Expansions for Atomic Ionization

We show that a frequently used nonperturbative approximation for atomic ionization rates is canceled out when corrections are taken into account. This explains the strong gauge dependence of previous results. A convergent and gauge-invariant expansion is obtained. Numerical results show that its first term, which may be calculated analytically in many cases, describes very well the time-dependent behavior of the ionization probability for very strong fields.