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Numerade Educator

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Problem 8 Easy Difficulty

Test the series for convergence or divergence.
$ \displaystyle \sum_{n = 1}^{\infty} (-1)^n \frac {n^2}{n^2 + n + 1} $

Answer

diverges

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Video Transcript

and this problem, we are given an alternating Siri's, and we have to determine using some tests that we have if this Siri's is convergent or divergent. So let's first review what we're given were given the Siri's where n equals one to infinity of negative one to the end and squared over and squared, plus and plus one So we can clearly see that this is an alternating Siri's. So what are we going to dio? We're going to take the limit as an approaches infinity of a seven. So what is that? That is basically the stuff inside are some, So this will make more sense. Once we start plugging things into our limit, we'll take the limit as an approaches infinity of negative one to the end times and squared over and squared, plus n plus one. Now this looks like a normal limit that we can solve. What should we do first? Well, let's essentially divide the numerator and the denominator by and squared We'll get the limit is an approaches infinity of and squared over and squared, divided by end squared plus n plus one all over and squared. And some terms we're going to cancel and make those limit a little bit easier. We'll get the limit. Is un approaches infinity of negative one to the end, times 1/1, plus one over n plus one over and squared. Now how would we solve this? Lemon with a little bit of a trick question, this limit doesn't exist. And why is that? Because this limit alternates between negative one and one. When n Assad or and is even, and that's obviously respect. Respective n is odd. It would alternate with negative one. If that is even, we would get one. So this is an alternating Siri's, which means that this is going to diverge. Our Siri's will diverge. It's essentially oscillating between two values forever. So I hope that this problem helped you understand a little bit more about alternating Siri's and how we can use a test to determine its divergence or its convergence