Application of schwarz's inequality to perturbation theory

It is shown by use of the variation theorem, that under fairly general conditions the energy given to an odd order of approximation by perturbation theory is an upper bound to the correct energy. Under less general conditions it is shown that for a positive definite perturbation the energy given by an even-order approximation is a lower bound and for a negative definitive perturbation it is again an upper bound. Some applications are discussed.