Applicability of perturbation theory

[CAF123]

Consider some system in some initial state $|k^{(0)}\rangle$. The probability that such a state makes a transition to some other state $|m^{(0)}\rangle$ can be computed to various orders in time dependent perturbation theory.

E.g the total first order probability that the system has made a transition from the initial state is $$P^{(1)}(t) = \sum_{m \neq k} p_{mk}^{(1)}(t)$$

For the perturbation theory to be valid, one requires that $P^{(1)}(t) \ll 1$. Can someone explain why this is the case? Is it something to do with the fact that we require all the corrections to the transition probability to be small so that when we sum all further contributions, they converge to 1? (Assuming we also include the probability that the state does not transit in time $t$ and then the transition probability will be 1 so that the system is definitely in one state after time $t$)

Thanks!

 

[HomogenousCow]

If the first order contribution alone is of significant order then the whole approximation scheme is unjustified.