Consider the series
Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.
The series diverges.
Proof. We use the limit comparison test with the series . Let
Then, taking the limit we have
(The final equality follows from Example 3 on page 405 of Apostol, which tells us that this limit is 1.) Therefore, and either both converge or both diverge. But since
diverges we can then conclude that
diverges as well