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