We consider a new method of series summation that generalizes the definition of sum and extends summations to the case when the upper limit of summation is less than the lower, or even negative. Building upon the method, we establish several rules of operations with divergent series. We evaluate at negative integer arguments the Riemann's zeta function and some other Dirichlet series in an elementary fashion. The application of the method to some other related zeta functions and to Dirichlet L-series is also discussed.