Consider the series

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

The given series diverges.

*Proof.* To show this series diverges, we first consider the series

This series diverges by the comparison test since

and we know

diverges. Further more the series

converges by the Leibniz test since it is monotonically decreasing and has limit 0. But then we must have

divergent. (Since if it was convergent, then adding it with the convergent series would converge, but we showed this sum diverged