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Determine whether the series converges or diverges.

$ \displaystyle \sum_{n = 3}^{\infty} \frac {n + 2}{(n + 1)^3} $

The series converges

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Oregon State University

Baylor University

University of Michigan - Ann Arbor

Boston College

let's determine whether the Siri's converges or diverges. We could use either the comparison test here or the limit comparison. Let me just go ahead and use usual comparison test. So this Siri's that's given. I can go ahead and replace this with an upper bound. So for my upper bound, if we look at the numerator first, let's go ahead and replace this with something so I could replace this with, Let's say, end Plus two in since two is less than or equal to to end and then I could just simplify that to get three. And so if I make my numerator or larger, that just makes my fraction larger. And then I'll just replace the denominator within cubed so here and is less than and plus one. And this implies and cubed is less than n plus one. Cute. So we made the numerator that and read larger. We made the denominator in green smaller, and both of those steps as a whole make that fraction. Let's do that in blue. The fraction is a whole got larger. This is why we have this inequality here. Now we have a simpler looking expression. Let's cancel one of those ends. So we're left over with this sum from n equals three to infinity ofthree over and square. Now, this is what we call a piece Erie's using the terminology from section oven point three. And then, in this case, we can use the pee test, which basically says that you have conversions if and only if P is bigger than one. In our problem, we have p equals two. That's bigger than one. So a convergence by it I mean this piece areas that we're looking at now we'LL use comparison test. I also know that in our in our original Siri's that our term and plus two over in place one cute. These are all positive numbers. This is what allows us to use the comparison test. So by comparison, our series also converges That's from n equals three to infinity and plus two over and plus one Cute. And that's our final answer