# Calc6.51

From Example Problems

Discuss the convergence or divergence of the series .

To show convergence for this series we will use Dirichlet’s test. The terms of this sequence are the product of and . The first sequence of numbers decreases toward 0 as . For the second sequence of numbers, , let us look at the partial sums

The terms of the sequence are {1, 1, 0, 0, 1, 1, 0, 0, ...}. Thus, this second sequence of numbers has bounded partial sums. Since our series has terms which are a product of the terms of one sequence which decreases to 0 and another which has bounded partial sums, by Dirichlet’s test, this series converges.