Consider the series
Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.
The series converges absolutely.
Proof. We know from the previous exercise (Section 10.20, Exercise #30) that the series
converges absolutely. Using the limit comparison test we have
Now we consider these as functions of a real-variable and make the substitution , and use L’Hopital’s rule three times to take the limit,
Therefore, since converges absolutely we have established the absolute convergence of