 In this video, we provide the solution to question number 10 for practice exam number four for math 12-20. We're given the series where we take the sum from n equals one to infinity of the sequence three n minus one over n plus one, and that entire term is raised to the nth power there. So we have to decide whether this series is absolutely convergent, conditionally convergent, or divergent, and we have to show all of our work and make sure we mention any convergence test that we're doing here. This could be a potentially easy one. I could potentially hard one. Now for this specific one, it turns out that the root test I think would be the easiest tool to use here because notice how we have a sequence raised to the nth power. The root test will be extremely advantageous in that setting. So if you take the limit as n approaches infinity of the nth root of this thing, for which we get three n minus one over n plus one, all raised to the nth power. So then the nth root is gonna cancel out the nth power. And so this limit then becomes the limit as n goes to infinity of three n minus one over n plus one. So this is a ratio, a balanced ratio. The leading term on top is three n. The leading term on the bottom is n. So this thing is gonna converge off towards three. So we get that the limit of this thing is three. Notice that three is greater than one. And so this tells us that the series is divergent by the root test. So we should be specific about that. It's divergent. Do make sure you circle the divergent above here, but it's divergent by the root test.