Secular Terms in Dyson Series to All-Orders of Perturbation

In classical and quantum systems, perturbation of an evolution equation is often invalidated by secular terms which diverge at late times. The diverging behavior of evolution can be remedied by various techniques of resumma- tion such as renormalization group or multi-scale analysis. In this paper, we prove that, in a generic quantum mechanical system, secular terms can be systematically removed to all orders in the Dyson series by the method of improved (renormalized) perturbation. A recurrence relation to provide an explicit method to remove the secular terms is given. As a byproduct, we give a simple method to obtain energy eigenvalues and decay rates to all orders of perturbation.