Consider the series

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

The given series converges conditionally.

*Proof.* First, we have

Furthermore, the sequence of terms is monotonically decreasing. We can see this by looking at

This derivative is negative for all (since the numerator is negative and the denominator is always positive). Hence, the sequence is monotonically decreasing. Therefore, by the Leibniz rule we have

converges.

This convergence is conditional since

This series diverges by the comparison test since

and

diverges

very clever!