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Use an appropriate test to determine whether the series given below converges or diverges_In n)2n =Select the correct choice below and; if necessary; fill in the an...

Question

Use an appropriate test to determine whether the series given below converges or diverges_In n)2n =Select the correct choice below and; if necessary; fill in the answer box(es) to complete your choice.0A The Limit Comparison Test withshows that the series diverges.0 B. The series diverges because the limit used in the nth-Term Test is (Type an exact answer:) Oc The Limit Comparison Test with 8 shows that the series converges.

Use an appropriate test to determine whether the series given below converges or diverges_ In n)2 n = Select the correct choice below and; if necessary; fill in the answer box(es) to complete your choice. 0A The Limit Comparison Test with shows that the series diverges. 0 B. The series diverges because the limit used in the nth-Term Test is (Type an exact answer:) Oc The Limit Comparison Test with 8 shows that the series converges.



Answers

Use the Comparison Test to determine if each series converges or diverges.
$$\sum_{n=2}^{\infty} \frac{n+2}{n^{2}-n}$$

Comparison test and too big the right a serious here we would take the maximum on the talk which it's and square and the maximum Bella in the bottom which is in military. So we should get in. Cochin is three under from one of the end and isn't is divergent. Do Trude. Uh that by the hammering Siri's here and now what? You need to turn the limit under and goes to infinity one of the and the violin binder and times and this one running by the end square bless one times with him and this one to coach. It'll emit ingress to infinity. We have here ends grab less one times a minus one devalued by and square times and press one So we see the maximum on the time. He's about three and I'm sort of sim power in the apartments of Victory and their final day Mr Iko, do the governor in front of maximum power which is going on the top and work in the bottomless well and he could you one here and one spirited and zero. Therefore, by the limit comparison test, we can prove that this race here is divergent

So if we call this a seven and he said, then one over and squared and we seek the limits as Angus, 23 of a Sedona peaceably, we get and squared over a squared vice and sign him. I'll be and over and Linus sign, and this limit is one. And since it's a constant and we know that, he said and converges, we know that a seven verities.

All right. We went to test convergence of both the sequence and the series. When we are given that a suburban equals two plus negative one to the end. Well noticed that if we start writing out some terms of the sequence When we plug in one we get one when we plug in to we get three. When we plug in three we get one and we plug in to we get three, we get one, three and so on. Every odd number results in one. Every even number of results in three. So the sequence keeps bouncing back and forth between one and 3 Endlessly. It keeps bouncing back and forth between the two, which means that the sequence does not have one unique limit. Therefore the sequence diverges now, how about the series? Well, since the sequence itself doesn't even converge, since it diverges that in particular means that the sequence Does not converge to zero. The reason that's important is that the end term test since that limit, since It does not approach zero, the end term tests tells us that the series also diverges

All right, we have that a sub N equals and over one plus n squared. And we want to test convergence of both the sequence and the series for the sequence. Well, since this is a fraction of polynomial where the degree of the numerator is smaller than the degree of the denominator, we can see that as N goes to infinity. This fraction is going to have to go to zero or to show it more formally. We can rewrite this by multiplying the top and bottom By one over N. To get that a sub N equals one over one over N plus N. And now we can more easily see as N goes to infinity, one over end goes to zero but N goes to infinity. So the entire denominator goes to infinity while the numerator is one. So The fraction has to go to zero. And we can say that the sequence converges Okay. As for the series uh when it comes to matters of convergence and divergence, typically the only thing that matters are the highest powers of end in the numerator and denominator. When we're looking at a fraction where there's nothing but powers of end, it's typically only the highest power terms that matter. So this suggests that the sum of a suburban behaves like the sum of an over and squared, which equals one over N. So, so if we let B sub n equal this, it suggests that the sums of s abandoned visa and behave the same. This is something we can check with the limit comparison tests. So we'll look at a suburban over B7 which equals and over one plus and squared times and over one equals And squared over one plus and squared. And as n goes to infinity, this ratio of n squared over one plus n squared. Well, here the degrees are the same. So as n goes to infinity, it's just the ratio of the leading coefficients 1/1, which is one. And the key here with the limit comparison test is that this limit one is greater than zero and it's not infinity, which means that the sum of a suburban and the sum of Visa Ben have the same behavior well, which means they either both converged. Both diverge, but the sum of B sub N is just the harmonic series, which means it diverges. So by the limit comparison test, the sum of Osama bin diverges by limit comparison. So in conclusion, although the sequence converges, the series diverges.


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