Consider the series

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.

The series converges conditionally.

*Proof.* We use the Leibniz rule. First, we have

since as . Furthermore, since is monotonically increasing, is monotonically decreasing. Thus, by the Leibniz rule the series converges.

This convergence is conditional since

Then, we use the limit comparison test with the series . We have

(Again, I’m pretending I can use L’Hopital’s on the integer valued functions, which I can’t. Technically we need to consider the equivalent real-valued functions, take the limit, and then return to the integer-valued functions.) Thus, diverges since diverges