then converges. Proof of Abel's test. Summation by parts gives where a is the limit of . As is convergent, is bounded independently of , say by . As go to zero, so go the first two terms. The third term goes to zero by the Cauchy criterion for . The remaining sum is bounded by by the monotonicity of , and also goes to zero as . Using the same proof as above, one can show that if the partial sums form a bounded sequence independently of ; (so that the sum goes to zero as goes to infinity) then converges. In both cases, the sum of the series satisfies: Summation-by-parts operators for high order finite difference methods A summation-by-parts (SBP) finite difference operator conventionally consists of a centered difference interior scheme and specific boundary stencils that mimics behaviors of the corresponding integration-by-parts formulation. The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique. The combination of SBP-SAT is a powerful framework for boundary treatment. The method is preferred for well-proven stability for long-time simulation, and high order of accuracy. See also Convergent series Divergent series Integration by parts Cesàro summation Abel's theorem Abel sum formula References Bibliography Summability methods Real analysis Lemmas in algebra