  We present an analysis of the adiabatic approximation to understand when it
applies, in view of the recent criticisms and studies for the validity of the
adiabatic theorem. We point out that this approximation is just the leading
order of a perturbation series, that holds in a regime of a perturbation going
to infinity, and so the conditions for its validity can be only obtained going
to higher orders in the expansion and removing secular terms, that is terms
that runs to infinity as the time increases. In this way, it is always possible
to get the exact criteria for the approximation to hold.
