Do you understand the logic behind the book's argument here?
You have to show (n+1)/((n+1)^3+1) < n/(n^3+1) if you want to use the alternating-series test.
That's the definition of absolute convergence. You need to know that so when the term comes up, you know what's being talked about.
Now look in the book for theorems that apply to absolutely convergent series to see why it might apply to this problem.