Consider the series
Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.
The given series is conditionally convergent.
Proof. By the Leibniz rule we know that if is a monotonic decreasing sequence with , then the alternating series converges. In this case we let
Then, is monotonic decreasing for . We can see this by considering
This is negative for ; hence, is decreasing for . Since is the sequence of the values of on the integers we have is decreasing if . Further, we have
Thus, we write,
This converges since the first term is some finite number (since it is a finite sum) and the second term converges by the Leibniz rule; hence, the sum of these two terms converges.
Finally, the convergence is conditional since
Letting we have
Hence, by the limit comparison test (and the fact that diverges) we conclude know that