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JH
Numerade Educator

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Problem 42 Easy Difficulty

Give an example of a pair of series $ \sum a _n $ and $ \sum b_n $ with positive terms where $ \lim_{n \to \infty} (a_n/b_n) = 0 $ and $ \sum b_n $ diverges, but $ \sum a_n $ converges. (Compare with Exercise 40.)

Answer

$a_{n}=\frac{1}{n^{2}} \quad b_{n}=\frac{1}{n}$

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Video Transcript

Let's give a example of a peer of Siri's for Anne and Bien with positive terms. So Anne and beyond are bigger than zero, and the limit of the fraction must be zero. So will eventually have to deal with this. And then we want that this convergence and that this Siri's diverges so here as an example, let's take being to just be one for each end and let's take and to be won over and square. Then this limit I can rewrite. This is the limit of one over and square over, just one. So that's the limit. One over and square. That's zero so that satisfied. So check also that the are positive terms, so that satisfies this condition. This Siri's converges. So here you can use the pee test with P equals two, and that's bigger than one. So convergence and then the sum of the B in this diverges because it fells the diversions test. What if you take the limit of bien? You just get one, and that's not equal to zero. And that means that the Siri's diverges and so we have all the conditions we want. Let's double check those is positive terms as here, the limit of an over being a zero Check the sum of the being diverges check, and the sum of the conversion is check. It's so this is an example that works, and that's the final answer.