Dirichlet's Test - Proof

Proof

Let and .

From summation by parts, we have that .

Since is bounded by M and, the first of these terms approaches zero, as nā†’āˆž.

On the other hand, since the sequence is decreasing, is positive for all k, so . That is, the magnitude of the partial sum of Bn, times a factor, is less than the upper bound of the partial sum Bn (a value M) times that same factor.

But, which is a telescoping series that equals and therefore approaches as nā†’āˆž. Thus, converges.

In turn, converges as well by the Direct Comparison test. The series converges, as well, by the Absolute convergence test. Hence converges.

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