Proof
Let and .
From summation by parts, we have that .
Since is bounded by M and, the first of these terms approaches zero, as n→∞.
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 n→∞. Thus, converges.
In turn, converges as well by the Direct Comparison test. The series converges, as well, by the Absolute convergence test. Hence converges.
Read more about this topic: Dirichlet's Test
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