Discuss the convergence or divergence of the series where .
It is well known that as , though it travels very slowly. So, we know that also goes to , though much more slowly. Taking the ratio of the two, decreases to 0 as because the bottom increases much more quickly than the top.
Now, looking at the sequence of partial sums of
we get the sequence of numbers (notice that k starts at 3). Since this sequence is always between 0 and 7, inclusive, it is bounded. Thus, our series converges by Dirichlet's test.