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Problem #3 For each of the following series , determine whether the three conditions of the alternating series test are satisfied_ If the answer is 'YES' ...

Question

Problem #3 For each of the following series , determine whether the three conditions of the alternating series test are satisfied_ If the answer is 'YES' , calculate the sumn of the terms up ton = 20 and estimate the 'tail' of the series Zac-21 Un. (_1)7 (-1)n (1)Z a) C b) c) (n? +42 n + Inn Z2nln n=l n=2

Problem #3 For each of the following series , determine whether the three conditions of the alternating series test are satisfied_ If the answer is 'YES' , calculate the sumn of the terms up ton = 20 and estimate the 'tail' of the series Zac-21 Un. (_1)7 (-1)n (1)Z a) C b) c) (n? +42 n + Inn Z2nln n=l n=2



Answers

(a) Decide whether the following series is alternating: \[ \sum_{n=1}^{\infty} \frac{\sin n}{n^{3}} \] (b) Use the comparison test to determine whether the following series converges or diverges: \[ \sum_{n=1}^{\infty}\left|\frac{\sin n}{n^{3}}\right| \] (c) Determine whether the following series converges or diverges: \[ \sum_{n=1}^{\infty} \frac{\sin n}{n^{3}} \]

Okay, so this time we have a value that is positive. So when ever we plug this in, we're always going to get a positive value. So in this case, we don't have an alternating Siri's.

Yes. At this time we have an A value of negative to This is a negative number. So when we played this end, we get negative two plus for an N plus negative eight. So that's noticed that we have minus plus and then minus. So we see that these sides are changing. In this case, we do have an alternating Syria.

So we want to determine if the following is an all alternating Siri's. So let's not have a minus sign and our next next term has a plus time and then we have a minus. In this case, we have an alternating Siri's.

In discussion. We need to use the ratio test to read your mind whether the evidence is converges. So first of all, let's discuss the ratio test samples submission. Okay it because to one to infinity kiki is a cities of positive terms. Such a bed limit and tends to infinity A. And plus one upon A. And it recalls to L. So if L. is less than one so we can conclude the cities is convergent. The second possibility is yeah if L is greater than one or if L is equals to infinity the series is divergent. And the third possibility is if yeah Ellis recalls to one. The ratio test his in conclusive. It means we cannot decide whether the city's converges or diverges. No we have the cities submission. And if he calls to want to infinity thank you 2 to the power and plus three upon 7 to the power and -1. And the series is a series of non zero bones. Therefore the value of A N plus one will be called to And plus one to the power three Into 2 to the power and plus four upon seven to the power and The two to the power and plus four will be calls to 2 to the power and into 2 to the powerful. Therefore A N plus one will be close to and we can take out in CUba as a common. So this will be calls to in CUba in 21 plus one upon end to the power three and two 2 to the power and into two to the power for upon 7 to the power and similarly the value of A. N. Will be called to thank you Into totally power and plus three upon 7 to the power and -1. The value of two to the Power and plus three will be calls to to to the Power and into two to the power three. And the value of seven to the Power and minus one will be calls to seven to the power and into seven to the power minus one. So this will be called to seven to the Power and upon seven substitute these values in A. N. So we get A and he recalls to seven into and cube into two to the power and into two to the power three upon seven to the power and no the ratio of A. N. Plus one upon A. N. Will be calls to and cube and two one plus one upon end to the power three into two to the power and into two to the power four upon 7 to the power and upon seven and cube into totally power three in two totally power end up on 7 to the power end. And this will be close to thank you bye into one plus one upon and to the power three into two to the power and into two to the power four upon 7 to the power and into 7 to the power and upon seven and cube into to Cuba in two totally power and no thank you will be cancelled by the uh Thank you. 2 to the power end will be cancelled by two to the power and 7 to the power and will be cancelled by 7 to the power and and To Cube will be cancelled by two Cube. So finally we have two into one plus one upon end to the power three upon seven. Now this ratio can be written as limit and tends to infinity. Mm and plus one upon a N. He recalls to limit and tends to infinity two by seven. Yeah one plus one upon and to the power three. So there will be calls to two by seven limit and tends to infinity one plus one upon end to the power three now substitute infinity on the place of end. So we get to buy seven into one plus one upon Infinity to the Power three. So this will be equals two two by seven Which is less than one. Since The limit is less than one. Therefore by the ratio test it can be concluded that the city's Submission and is because to one to infinity And Cube into 2 to the power And plus three upon 7 to the power and -1 converges. So this is the final answer for this problem. I hope you understood the solution. Thank you


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