Assume that bₙ is a sequence of positive numbers converging to...

Question

Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.

a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]

Accepted Answer

Let a subn be a sequence of positive numbers converging to 1. Does the series sum n equals 1 to infinity of 2 a subn divided by n6 to the n converge or diverge? We have two possible answers, which are converges or divergence. Now, to solve this, we'll make use of the limit comparison test. This tells us that for positive sequences, AN and BN. If l equals a limit as in approach affinity of a subn divided by bin exists. And l is between 0 and infinity, then the series for a subn and the series for bn both converge or both diverge. In particular, if L equals 0 and B converges, then A converges. Or if L equals infinity and B diverges, A also diverges. Otherwise, this test is inconclusive. Now, to solve this, we will identify a comparison series. Our series currently is 2AN divided by N6 raised to the end. Now this does tell us that a subn converges to 1. So, we can say this is 2, divided by N, 6 raised to the N. From here we can choose piece of it. Now BN, we can compare this to something easier to use. In this case, we'll say 1 divided by 6 to the n. This is a known geometric convergence. So, From here. Let's plug these into L. We know l is the limit. As n approaches infinity of a n divided by b n, which is the limit as n approaches infinity. Of 2 divided by N 6 to the N divided by 1 divided by 6 to the end. Now this then simplifies. We just end up getting 2. Divided by n With our limit Now this limit is equals to 0 because we'd get 2 divided by infinity. So by the limit comparison tests, if our limit equals 0 and BN converges, then AN should also converge. Now, I did mention that BN is a convergent geometric series. This is because we have 1 divided by 6 raised to the end. Where r is equals to 1/6, which is less than 1. That means that converges. So, by the alternating series tests, because our limit is 0 and BN converges, AN also converges. That means our series converges, which means our answer is answer A. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]

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Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.
a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]