The time-dependent quantum perturbation theory developed by Born, Heisenberg and Jordan in 1926 is
revisited. We show that it not only reproduces the standard theory formulated in the interaction
picture, but also allows one to construct more accurate approximations if time averaging techniques
are employed. The theory can be rendered unitary even if the expansion is truncated by using a
transformation previously suggested by Heisenberg. We illustrate the main features of the procedure
on a simple example which clearly shows its advantages in comparison with the standard perturbation
theory.