Consider the series

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

The given series diverges.

*Proof.* The series diverges since the limit

(Since the limit of the even terms is 1 and the limit of the odd terms is

Why are you considering the (-1)^{n} when taking the limit? Shouldn’t you just consider a_n=\cfrac {1}{n^{\cfrac{1}{n}}}?

That’s what I thought, though I think you can use theorem 10.6 since there’s no mention about the positivity of “an”, and in this case “an” goes to -1 or 1 (not zero)