Dirichlet’s and Abel’s Tests

We can now use Abel’s partial summation formula to establish a couple other convergence tests.

If is a sequence whose sequence of partial sums form a bounded sequence, and if is a decreasing sequence converging to zero, then the series converges. Indeed, then the sequence also decreases to zero, so we just need to consider the series .

The bound on and the fact that is decreasing imply that , and the series clearly converges. Thus by the comparison test, the series converges absolutely, and our result follows. This is called Dirichlet’s test for convergence.

Let’s impose a bit more of a restriction on the and insist that this sequence actually converge. Correspondingly, we can weaken our restriction on and require that it be monotonic and convergent, but not specifically decreasing to zero. These two changes balance out and we still find that converges. Indeed, the sequence converges automatically as the product of two convergent sequences, and the rest is similar to the proof in Dirichlet’s test. We call this Abel’s test for convergence.