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Test the series for convergence or divergence using the Alternating Series Test:(-1)" - Je5 / nIdentify bEvaluate the following limit:Iim n-ISince Iim n-1and b...

Question

Test the series for convergence or divergence using the Alternating Series Test:(-1)" - Je5 / nIdentify bEvaluate the following limit:Iim n-ISince Iim n-1and bfor all n, the series diverges by the Alternating Series Test

Test the series for convergence or divergence using the Alternating Series Test: (-1)" - Je5 / n Identify b Evaluate the following limit: Iim n-I Since Iim n-1 and b for all n, the series diverges by the Alternating Series Test



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.

Hello. So here we are going to be using the ratio test. So we have a seven is equal to the natural log of N over two to the end. And then by the ratio test we have varied the limit as N goes to infinity of while. Hmm Plus one over a sub N. So we have the limit as N goes to infinity of well um the natural log of M plus 1/2 to the N plus one times two to the end over natural log event. That gives us the natural log of N plus 1/2 times the natural log of end. So, um we'll hear the limit right, May not be, it's not clear. So we can go ahead and use low petals row. We take the derivative of the top and divided by the river at the bottom. So then we get um the limit sure as N goes to infinity. Well, I love potatoes, roll this is just going to be equal to um and over two times and plus one is, and it goes to infinity. On the limit here is just equal to one half, so since we have that one half is less than one, um that tells us that the given series con vergis keep.

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.

In this question we will reveal about telescoping three tests son. That early scoping. Sorry I asked. And this tax here it has a special form where the submission here will be between the I. N minus the I. N. Plus one on the vice versa. Uh And we try to expand this series here like an echo. What you want to infinity. We will see that I one minus 82 and then plus eight two months +83 and plus eight three months +84 and so on. We will see that we will have the sequence of cancellation here And at the end we have left with the term in the first few times. So particularly we have the question of some mission one of them and just one -1 other and plus two. So we noticed that the form of this one will be the same from on this one. There are only the different um there are only they're different in the increment first will be endless one. The only one will be endless to now the technique to do this problem will be we will try to expand this series here. Start from one. So we will try to start with one first it will have to end equal to one and then any CO 22 and 23 and go too far and so on. So from the case n equal to one whose again the 1/2. 1 is one of three. This will be plus because it is a series now for an equity too. We have won over three minus 1/4 17. For any co +23 we have one hour four minus 1/5. And for any what you found we have one of five minus 1/6 and continue to the and um here will be one of the endless one. Man is one of them and blessed you and then continue to forever. Now the team will be up and on the market. This one we cancelled and with this this one we canceling with this, this one we cancel out with this and so on. So we see this one will be canceled out as well. So we will come up, comes up with the limit and get to infinity and we have left in the first time will be one half And this term here will be -1 other and this jew and notice that as anger to infinity this term here goes to zero. Sorry there's only been one of the two. So we have left with only the 1/2 and isn't exactly the same. Okay. Yeah. So their father doesn't imply stand this reasoning would be unfortunate


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