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.
Think you have a mistake:
If B_k is decreasing, then:
B_{k+1} – B_k < 0.
So it is not the case that:
|A_k(B_{k+1} – B_k)| <= M(B_{k+1} – B_k)
Sorry, yes, the indices are swapped.