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Use the Direct Comparison Test to determine the convergence or divergence for these series_ Include all the criteria for test, show all the work and state conclusio...

Question

Use the Direct Comparison Test to determine the convergence or divergence for these series_ Include all the criteria for test, show all the work and state conclusionA) n= 2" +3B) n-2 n=3

Use the Direct Comparison Test to determine the convergence or divergence for these series_ Include all the criteria for test, show all the work and state conclusion A) n= 2" +3 B) n-2 n=3



Answers

Determine convergence or divergence for each of the series. Indicate the test you use. $\sum_{n=1}^{\infty} \frac{n^{n}}{(2 n) !}$

Hello. So here we are going to be using the ratio test where are a seven is N squared over N factorial. Then for the ratio test we are going to be evaluating the limit as N tends to infinity of a sub N plus one over a sub N. So that's going to be and plus one squared over and plus one factorial and then times and factorial over and squared. Well this is gonna be just equal to the limit. Yeah. As N goes to infinity. Um after that we simplify here. This is gonna be equal to just M plus one over and squared. And as N goes to infinity, the higher power this goes to zero. So the limit here is equal to zero. And since we have that zero is less than one that tells us by the ratio test that our series con verges. Yeah. Damn. Okay.

Okay, so here we are going to be using the term divergence test. So here's our given series, we get that a seven is equal to 1 -1 over end to the end. Okay, so um what we can do well a common form of um so basically looking at one minus one over end to the end, this should remind us um of while E right, because while basically E to the minus one um is equal to Each of the -1 um basically is is equal to the limit as N tends to infinity of one minus one over end to the end. So since the terms of the series approach E to the minus one and not zero um therefore by the end of term divergence test right away we can say the series diverges because um right if a series converges then the series, the terms himself much approach zero terms don't approach zero, then the series must diverge. So here we are approaching each the minus one E to the minus one, um Is not equal to zero. So therefore, by the end of term divergence test, our given series diverges. All right. Take care.

Hello. So here we are going to be using the ratio test where we have a seven is equal to n squared plus 1/3 to the end so that the ratio test subjects. We take the limit as N goes to infinity of a sub N plus one over a suburban. So that is then going to give us well it gives us an plus one squared plus 1/3 N squared plus three and as N goes to infinity, the limit here is equal to one third. Now since we have that one third is less than one. Therefore the ratio test tells us that the given series con purchase. Yeah. Yeah. Yeah.

Okay, so here we are going to be using the limit comparison test where our ace event is going to be equal to n plus three over n square times the square root event. Then we're going to compare this series um which with a similar series which should be similar for large end. So if this is our ace event, we're going to compare this with the Suburban which is going to be equal to let this happen to be equal to one over N times the square event. So now using the limit comparison test, we take the limit as N tends to infinity of well a cement over um piece of end so of a suburban over B seven. That gives us N square times square event plus three N times square event. All over. N square times square event. Which is going to be equal to the limit as N tends to infinity of just one plus three over end or three over in um goes to zero. So therefore this limit here is equal to one. Um And since we have that while zero is less than one, um we get that both of this series are either going to converge or diverge together, and here we have the Beast event converges because Beast Event one over end times square to bend um can be written as one over end to the three halves, so that's a P. Series where P is equal to three halves, So therefore be seven converges. Therefore our given series con verges as well, jesus.


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