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22 [2r3 Points]DETAILSPREVIOUS ANSWERSSCALC8 11.5.007.MI:Test the series for convergence or divergence(-1)" (SnEvaluate the following limit: (If the quantity d...

Question

22 [2r3 Points]DETAILSPREVIOUS ANSWERSSCALC8 11.5.007.MI:Test the series for convergence or divergence(-1)" (SnEvaluate the following limit: (If the quantity diverges, enter DIVERGES.) (-1)" (Sn lim(Sn Since Iim (~1)" doos not exist 7n + 4Ihe sones I8 divergentNeed Help?Rpd luLakahbMtesn

22 [2r3 Points] DETAILS PREVIOUS ANSWERS SCALC8 11.5.007.MI: Test the series for convergence or divergence (-1)" (Sn Evaluate the following limit: (If the quantity diverges, enter DIVERGES.) (-1)" (Sn lim (Sn Since Iim (~1)" doos not exist 7n + 4 Ihe sones I8 divergent Need Help? Rpd lu Lakahb Mtesn



Answers

In Problems $15-28,$ use the Limit Comparison Test to determine whether each series converges or diverges. $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}+1} $$

Comparison tests to determine if this versions of coverages repair this series too in order to to the Cape. You know that this courageous by the geometric serious since by the geometric series, the episode value of our would be less than once. It's it's equal to one half so that we take That as our B seven. How we're going to take the limit has and approaches infinity Whatever to to the end plus one over Number 2 to the end. So that gives us the limit has an approaches infinity to see the end over 23 N plus one that equals one. Therefore, since one of the series converges, both converged by the limit comparison tests.

Comparison tests determine whether this series converges or diverges. So this is going to be our and here and I'm going to have the summation of one of our case squared. Pr bien. So, okay, we definitely know that this converges my P series. It's P is equal to two which is greater than one, right? So then let's go ahead and check that out. The limit has approaches infinity uh huh. Whatever. And plus one. I'm Sam Plus two. Going to pay one over N squared. So taking its reciprocal here simplifies to and squared over and plus one times one plus two gives us hand squared plus 200% plus two. And so that limit as N approaches infinity equals one. And so therefore, since we know that one of our case squared converges than both of them converge. So, since it equals two, the number and Britain.

Comparisons has to determine if this serious coverages for diverges compare that to five over read to the cake. That's going to be RPS so bad. You know that that by the geometric series convergence since one third system one, It's 1/3 to the Cape. And so then we take the limit as and approaches infinity Of 5/3 to the end. Plus two. Try to buy 5/3 to the end. Simplifying that out here, we get the limit has and approaches infinity of five times 3 to the 4th power. Try to buy five times 3 to the n plus 10 equals one. Therefore, since we know that one of them converges, both can approach.

Doing a comparison tests to determine whether the serious can purchase or temperatures on the safe side bend when you compare that to run over kiss. There were four K. And so this you know, do I purchase by P serious? It's P is equal to one And any time it's equal to one or less than one. Its star Brigid. So that. Alright, so that's going to be our B- seven. Take the limit as N approaches infinity Of and over two K. I mean to end That is one squared mm Divided by 1/4. Okay, simplifying that out here to get floor and squared over. That's four and squared -4 hands plus one. Oh take the limit as an approaches infinity equals one. So therefore you know that since one of them starve urgent that vote for them, words have urgent.


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