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Does the following series converge or diverge? 0 (2n)l 2 n= 10"nl0 The series diverges. 0 The series converges 0 The series diverges absolutely 0 The series di...

Question

Does the following series converge or diverge? 0 (2n)l 2 n= 10"nl0 The series diverges. 0 The series converges 0 The series diverges absolutely 0 The series diverges conditionally The series converges conditionally

Does the following series converge or diverge? 0 (2n)l 2 n= 10"nl 0 The series diverges. 0 The series converges 0 The series diverges absolutely 0 The series diverges conditionally The series converges conditionally



Answers

Does the series converge or diverge? $$\sum_{n=0}^{\infty} \frac{2 n}{\left(1+n^{2}\right)^{2}}$$

Again this caution. The series given is submission. And it was 202 in finite One divided by any Squire plus two. n plus two. Okay. And be able to decide whether the series converges or diverges. Okay? So and is one divided by the square plus two. N plus two. Okay. And this behaves like one divided by any square. Okay? Because one divided by the square. Is there? Okay? And we can say it is also that one divided by and the Squire plus two N plus two is also less than one divided by any square. Okay, Less than equals to four. Some values. And now we know that submission and it calls to 12 in finite one divided by the square. Why I'm taking one because zero will be undefined. Okay, submission. And it was to 1.5 Night one divided by any square is can Virgin. Okay. And more than using P series at P equals to two. Okay. It's convergent. And that given series that isn't the question. Okay, submission. And it was 202 in finite one divided by any square plus two. One plus two is less than okay, less than Equals two submission and equals to 02 in finite one divided by any square. So the given citizens in the question also convergent. OK. So our answer will be can work jen's. Then what is? Thank you?

Okay. And discussion of series is given that is submission and equals to 02 In finite four divided by two N plus one. And were able to find out whether the series converges or diverges. Okay. And now in this series, N is four divided by two. N plus one. Okay. And suppose the N S A function F X. Okay. Where effects will be? So effects will be four divided by two X plus one. Okay. And this function Effects is a continuous positive and decreasing function. So we can take integral integral test to identify the convergence or divergence. Okay. And esper cure. Um 9.3 if the integration one point finite fx bx converges. Okay. And then cds converges. Okay. And if if this integral diverges then the series also diverges. Okay, so no problem. We will take this. I want to in finite F x D x. And it will be integration 12 and finite fx is four divided by two X plus one dx. Ok. So now we will do the part of integration and it will be integration first of all, four Outside. And it will be limited limit eight tends to in finite and we will take it 12 A and one divided by two X plus one B X. Okay. And now it will be four limit eight times two in finite. And the integration of one divided by two X plus one. It will be one by two. Okay. It will be one x 2. Ellen two x plus one. Okay. And the limit will be from one to and now We will take this one x two out of this. So it will be to limit eight tends to in finite and this will be and then two x plus a 128 And now it will be too Limit eight tends to infinity. Now we will apply the limit. So it will be Ln to a plus one minus Ellen, it will be two plus one, that is three. Okay, And now limited tends to zero so this tends to infinity. So this will become infinite. Okay to it will be infinite minus Ln three and now The whole person that is in finite -2 Ln three that can be written as in finite. So it means that our integral diverges. So I spent 39.3 over cities also diverges. Okay, so series diverges and this will be the final answer of discussion. Thank you.

Again this question of cities is given that is submission And it was 202 in finite. Two divided by one plus four and square. And we have to find out whether the series converges or diverges. So here I am will be two divided by one plus four and square. Okay. And we suppose A N equals to some function. FN So the value of effects, it will be two divided by one plus four X square. Okay. And here fx is continuous, decreasing and positive function. So we will do the integral test to find out whether it converges or diverges. Okay, so integration of F X D X 12 in finite it will be integration 12 in finite. Fx is too divided by one plus four X square dx. Ok. And what the Durham says, if this integral converges then the series converges and if the integral diverges then the series diverges. Okay, now we have to solve this integral. First of all it will be limit tends to infinite and we can take 12 and now two divided by one plus four X squared dx. Now we will take you outside and limit. It tends to in finite from one to A. And it will be one divided by one plus four X square can be returning to X. Holy square and dx. Ok. Now we will substitute U. Equals to two X. So do you will be two dx. So D X will be do you divided by two? Okay so either to limit A tends to in finite it will be 12 A One divided by one plus you square. And the exit will be do you divided by two? So do you divided by two? I'm taking outside. Okay. Now it will be to buy to cancel out and limit. Okay limit A tends to in finite and integration of one divided by one plus you square. It will be our then Our 10 you okay limit 12 A. And now We will back substitute the value of you. That is two weeks. So it will be limited. eight tends to in finite it will be our 10 two X. Okay. From one to and now we will apply the limits. So it will be our are time to weigh minus R 10 1. Okay. And now Limits eight times to infinite. So this will be our 10 in finite are 10 2 into infinite. That is also infinite minus R. 10 1. Okay. And 10 universe infinite. That is by by two and minus R. 10 1. Okay. And this is a definite value. So we can say our integral converges and our series also converges. And this will be the final answer of this question. Thank you.

Again discussion A series is given their taste some mission and equals to 02 in finite and two divided by the square root of two plus. And and we have to we have to find out whether the series converges or diverges. Okay. And in the cities am will be two divided by square root of two plus end. Okay. And we will suppose a N equals to some function effects. So effects will be two divided by the square root of two plus X. Okay. And this effect is decreasing continuous and positive. So we will do the integral test to find out the convergence or the divergence. Okay. So as part of your um 9.3 together integral test where integration of F X. D X. One point finite. If if this integration converges then the series A. M. Converges. Okay. And if this integration diverges, okay then the series also diverges. Okay, so this is the huron. We will apply. So integration from one point finite. If X D X. Or we can say integration want to invent it effects is here too divided by the square root of two plus X. And dx. Ok. And now we will check whether over our integral converges or diverges. So it will be to outside and it will be limit eight tends to in finite and it will be one to a Again one divided by square root of two plus X. Dx. And now it will be too. and the integration of one divided by square root of two plus access to to multiply by to multiply by two plus X. Okay. First of all, limit A tends to infinite here and it will be from one to a And now it will be too Limit eight tends to in finite. And when we put uh first of all this will take you outside. Okay. And now it's choir out of two plus x two square root of two plus a minus the square root of two plus one. Okay. And now It will be four limit eight and 25. Night it will be the square root of two plus a minus the square root of three. Now we will apply the limit eight times two in finite. So this will be in finite. That is four multiplied by and finite minus road three. And that will be in finite. Okay? So we can say our integral diverges and S 49.3 over integral diverges. So our cities also diverges. So the answer will be series diversities. Thank you.


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