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Find the series' interval convergenca and, within Ihis intorval the sum ol ihe sonos a5 fundion ol x.Eer"Find the series' inlerval 0f convergence. Se...

Question

Find the series' interval convergenca and, within Ihis intorval the sum ol ihe sonos a5 fundion ol x.Eer"Find the series' inlerval 0f convergence. Selecl the correcl cholce below and, necessary: hll in the answat box comolele your cholce. The Inierval Oi convergence (Type compound Inequality Use Integers Iracuons Ior any numbers the erprasslon ) The series converges Only at * = (Type an integer ftaclion )The series converges Ior all valuos 0f KThe sum ol tha sories 6 E6-0

Find the series' interval convergenca and, within Ihis intorval the sum ol ihe sonos a5 fundion ol x. Eer" Find the series' inlerval 0f convergence. Selecl the correcl cholce below and, necessary: hll in the answat box comolele your cholce. The Inierval Oi convergence (Type compound Inequality Use Integers Iracuons Ior any numbers the erprasslon ) The series converges Only at * = (Type an integer ftaclion ) The series converges Ior all valuos 0f K The sum ol tha sories 6 E6-0



Answers

Find the interval of convergence of the series and, within this interval, the sum of the series as a function of $x$. $\sum_{n=0}^{\infty}(\ln x)^{n}$

So our first objective is to find the interval of convergence and then after that find the some of the series that's within that interval. Go ahead and use the ratio test to find out what the interval of coverage is just going to be. So we'll have X -1 two and two. Our four Plus 1. We would multiply by it's reciprocal symptoms, dividing by what we started with. So we have four to the end over X -1 two and power. So that gives us So that's going to be the absolute value of X by this one Squared over four. We simplify everything To be less than one. So then we'll have listed for here X minus one Absolutely there off here squared and then we screwed both sides here Of X -1 being less than two of the value of So we have negative two is less than X. Place one just less than two. Then from here we'll add one to both sides. We have negative one less than X which is less than three. So then from here. Okay, so I could see that if we okay, so we try plugging in those endpoints that we have here to check them out. So negative one and 3 if we plug those in wind up with for Since the square will have four to the end over 4 to the end. And so in which case data, verge at the end points so our interval of conversions is going to be a negative one, A couple three. So this is your interval Okay, that from here to find the some of the series as a function of X and what we need to basically put in the form of a over 1- are here and our a one. So if we plug in zero in there we get Just one. So it's going to be equal to one over. It's an from here. That's So it's X -1 squared over four. The power here, So I have 1 -1 express one or four. Yeah squared here. Okay. All right. And so then just to simplify this a little bit further, let's multiply by 4.4. Top and bottom. So we'll have that the some of the series here, It's going to be able to 4/4 x minus one squared. And that's I believe that as a because that's a function of X ray there.

Fractions of this. So let's apply the ratio test. So thank God to in terms or it's too and to endless too. All over three times 5 times two N plus one tips to end three, all that times two X to two and three. And multiply that by what we had originally which was bye plus one Within over two times 4 to me, the multiplication to end and and divided by Next to the two n Plus one. Okay, so we have here is X squared and if we take the limit as N approaches infinity here. So this goes with that, that goes with this here and then 22 ends to ends here go to one to relax with the value of X squared less than one, Which means we have from negative one 21 Okay, so now let's apply the end points and check them out. So tax equals negative one. So you've got all of the summation of this two times four to win over three times 5, two and plus one And -1 to the end to end this one. Okay, So this actually would converge by all training series, whereas at X equals one, which has been the summation of two times four to hand three times five Times 200 Plus one. So be like that times one. So here, so this would actually diverge. Uh And that's by reason of by like diversions tests or Yeah, she doesn't think so. Then our answer is It's from negative one 21

Probably geometric series, you know, here you can see that horror is equal to 1.075, which is greater than one, therefore this series Virtus.


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