Consider the series
Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.
If then the series diverges, if the series converges conditionally, and if , then the series converges absolutely.
Proof. Case 1. Assume . Then
so the series must diverge.
Cases 2 & 3. Assume . The series converges since
Now, we can apply the Leibniz rule since (for ),
Further, the sequence is monotonically decreasing since
since , and for
Hence, the derivative is always negative, so the function is decreasing. Therefore, the series of terms taking this function value on the integers must be decreasing as well. By the Leibniz rule, this means the series converges for all .
Finally, we must show this convergence is conditional if and absolute if . So, we consider,
Considering the series we have
This means the two series will converge or diverge together. But, we know converges for and diverges for . Hence, the series in the exercise converges conditionally if and absolutely if