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Find the radius of convergence_ R; of the series _nlxh 2 . 5 . 8 (3n - 1)Find the interval, I, of convergence of the series_ (Enter your answer using interval notat...

Question

Find the radius of convergence_ R; of the series _nlxh 2 . 5 . 8 (3n - 1)Find the interval, I, of convergence of the series_ (Enter your answer using interval notation:)

Find the radius of convergence_ R; of the series _ nlxh 2 . 5 . 8 (3n - 1) Find the interval, I, of convergence of the series_ (Enter your answer using interval notation:)



Answers

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 8}^{\infty} \frac {(x - 2)^n}{n^2 + 1} $

But the serious districts of the farm expel and inviting by fire about any attempts and about five from one to infinity and I would you use the racial test on in the rest of tests? I need to compute the limit off. The I am is one over. I am industry infinity. And in this camp I am this one. So I should get the expert and this one of a five and this one. And now I am this 15 and now dividing by n. So I need to want to play by the reciprocal. So they're 5 10 and five anywhere. Expel! And now we see that we can consider the expo and we just Ah, 50 And with this power here and then we can would this one into the same about five. And we'll bring the X our side. I'm so this five outside and for we have now become it becomes the absolute of X over five times the limit off the have the big about five outside inside we had any other am this one and just infinity. And we know that we can push the limit inside about five and then we have the absolute X over five times the about five outside, and so have really limit. And of em, this one. And we see that this limit him. We called you one. Therefore eventually, in the absolute of X over five only energy me convergent this limit him a space monitor and one wasn't even. Only if absolutely, banks will be smaller than five on even only if X must be between minus five and five. Let me test the two and concertina. So when x echo Germany's five and then we shouldn't get now that series will become the we have a manus fi about and, um, fiber and and about 511 to infinity. And it becomes the minus one Bella and over in about five and isn't will be. Clearly we convergent by the piece is gonna be equal to five credit and one And now I'm so for the, uh, right and bond and go to five, we have a solution. Now we're becomes the one over in about five from one to infinity and listen again clearly that it will be convergent by the B E co 25 credit and one is a piece Aries. So we conclude under interval it will be four minus 5 to 5. Including that you in point. On the right is a culture five.

We'LL figure out our radius of convergence without taking any shortcuts would just use the ratio test here. So lemon is n goes to infinity of a n plus one over and where a And is this whole chunk here, including the X values? Okay, so this might get a little messy. So this is a limit as n goes to infinity Absolute value X to the n plus one divided by X to the n is just x Okay, And then we still have this in plus one factorial. And we still have this whole chunk happening down here one times three times that that that times two and minus one times two and plus one minus one for the A N plus one guy, we have to go all the way until two times in plus one minus one and divided by alien is something as multiplying by the reciprocal of Anne. So now we're going to be multiplying by one times three times, dot, dot, dot all the way up to times two in minus one, and we're going to be divided by in factorial. Okay, so this chunk here is going to cancel out with this whole chunk here, the two times in plus one minus one is going to survive and in factorial is one times two times three times. Not that that times in, we're going to get everything within factorial to cancel out with the in plus one factorial but the n plus one in the in fact plus one, factorial will survive. So this is going to be simplified to limit as n goes to Infinity Act's times in plus one. Nothing's going to cancel out with the M plus one here, and then nothing will cancel out with this two times and plus one minus one. So that's still going to be there. Okay, as N goes to infinity in plus one divided by two times in plus one minus one, that's going to be one half because this is, ah, linear term with leading coefficient one. This is a linear term, with leading coefficient to sew as n goes to infinity, the limit is one divided by two, which is one half So this is absolute value of X over to, and when we're trying to figure out where we get convergence with the ratio tests, remember, we want for this to be strictly less than one. Okay, so that means that absolute value of exes. Lesson two, the radius of convergence. It's going to be too right cause X is going to be trapped between minus two and positive too. The length of that interval is for radius of convergence is half of the length of the interval of convergence. Okay, so now we need to figure out whether or not we include minus two and whether or not we include too. Okay, so what happens if we plug in X equals to here? X is equal to two than we have in factorial times to the end divided by one times. Three times that. That that times two and minus one case. Remember what X R what in Factorial is in? Factorial is one times two times three times that that that times in two to the end as two times two times that. That that times two where we have in different copies of two here. So if we multiply in factorial by two to the end, we can do one times two, two times two three times to each one of the factors in this product can be multiplied by one of these. Choose here. And then we can write this as one times two, which is two two temps to which is four, three times, too, which is six that that that all the way up to to end. And we're dividing by one times three times. Doctor. At that. Yes. I should put a five here. So we have the dot that thoughts in the same place times two in minus one. All right. But now we just put parentheses in the right place. Then we can see that this is not going to converge because the terms are not even going to go to zero. So we just put Prentice's around each of the guys here, apprentices around to over one footprint of seas around former three that that that keep going until we have Prentiss is around two in over two and minus one. And notice that each one of these terms is bigger than one. So this whole thing has to certainly be bigger than one. So in particular, it can't possibly go to zero. So these terms do not go to zero. If the terms don't go to zero and we can't possibly get convergence. So we get divergence and we get the same type of thing happening. If we plug in X equals minus to plug in X equals minus two, we get the same type of thing. Except now we would have AA minus one to the inn happening here. But again, it's not going to change the fact that our terms are not going to zero. Okay, so we get divergence if we plug in two or minus two into R sum here. So both those values we have to throw out of our interval of convergence. So we leave that as an open interval here. So not included minus two and not including positive, too.

Case of the radius of convergence is going to be limit as n goes to infinity of absolute value of a N over and plus one where a Is this chunk here without the X value And yeah, maybe we'll well, we'll do it using the ratio test just in case this is confusing to anyone. So the ratio test would do the same type of thing Except we have the sub script with the implicit one of top now and buy this being now we do mean this whole chunk here, including the X values. Hey, so this is going to be X to the n plus one over one times three times that that that times two in minus one times two in plus one minus one. Okay, And then so that's just are being plus one. And we're dividing by being somewhat planned by the reciprocal. So now I have an X to the end over there. Enough top. We're gonna have one times three times that, that that times two and minus one. And this whole chunk here is going to cancel out with this whole chunk here and actually and plus one divided by X to the n is just going to leave us with X. So now we have limits as n goes to infinity of absolute value of X and nothing got rid of this two times in plus one minus one. So we still have that there and for using the ratio test to figure out where we get convergence. We want for this to be something less than one. But notice that this term is going to go to zero as n goes to infinity. So it doesn't matter what value we plug in for X. We're always going to end up getting zero here. Zero is certainly less than one. So X is allowed to be anything for the radius of convergence is infinity, Which is exactly what we would have gotten if we I just did this here. This will short cup would have gotten that R is equal to limit as n goes to infinity of two times in plus one minus one and that would be infinity. Okay, So if the radius of convergence is infinite than the interval of convergences, certainly going to be infinite as well

In this problem we want to work on finding radius is of convergence and intervals of convergence for a series. And so again, we're going to need this key idea here that we need to know the limit of the end plus first term over the end of term. When is that? Less than one? And then we need to check our endpoints individually. Okay. All right. So for this example, we're gonna take a look at this series and equal zero to infinity of negative one To the N -1 over N. Times 5 to the end X to the end. Okay. All right. So let's get started. Let's take a look at the limit of the N plus first term divided by the end of term. Okay, So it's going to be a big fraction numerator, is there is always an N plus one Instead of an end. So it's going to be negative one To the end plus one minus one Over N Plus one. 5 to the n plus one Times X to the end Plus one. All right, denominator of our big fraction we're looking at Is just the end of term. So negative one to the end -1 over and 5 to the end times X to the end. Okay. Alrighty. Let's do some simplifying. Let's flip that a bottom fraction and multiply it instead. Lemonis N goes to infinity. Okay already, so we're going to have negative one to the end plus 1 -1 which is just end. Okay, so doing a little simplifying right here, that is just going to be in Over and plus one 5 to the end plus one. Okay? And then flip and multiply the bottom fraction to simplify. So N times five to the end over negative one To the N -1. Okay. And then my ex to the end plus one and my ex to the ends. Those are going to simplify and just get me the X. Okay. Alrighty great. Now a little more simplifying here. 5 to the end and 5 to the N Plus one. Those are going to simplify And just give me a five And then likewise the -1 to the end and the -1 to the N -1. Those are just going to leave me with a negative one to the first power. Okay already, so to simplify this a little more we get the limit as N. Goes to infinity absolute value uh negative one, number five. Alright my negative one from here. My five from here The n. over n plus one. We write that like this And over n plus one. And then my ex. Okay next step take out everything that's not an end because it doesn't factor into the limit. I'm going to factor out The -1 5th times X. And I'm left with the limit as N. Goes to infinity Of the absolute value of an over N Plus one. Okay. Alrighty. Now again this limit we should recognize that that goes to one By local Charles Rule. And so I am left with the absolute value of -1 5th. See X Too long. Okay. And if this is driving you crazy this is the same as the absolute value of 1/5 X. Okay Because again the absolute value of -1 is just one. All right. Now by our key statement up here I will need that limit to be less than one for my series to converge. So I need That limit which turns out to just be the simple idea of 1/5 x and absolute values. I need that to be less than one. Awesome. Let's multiply both sides by five. So I get the absolute value of X is less than five. Okay. And this is my radius of convergence. Yeah, So my radius of convergence is five and then my start of my interval. Well I think I'm going from negative 5-5 but I don't know about the endpoints yet. So I have to check those. All right. And so that's what we'll do next in this number two here when we check the endpoints individually. Okay, so let's check those out. Okay, again, how we do that is we're just going to plug in five for the X. We rewrite that. Okay, so let's check when X is equal to negative five. Okay? So when X equals negative five, my series looks like this and equal zero to infinity negative one To the end -1 Over N. Times 5 to the end And then times negative five to the end. Okay. All right, so this one you have to pay attention. There's something uh really simple but subtle happening. Okay, if I do my crossing out, I can cross this out in this out and I'll be left with a negative one. Okay? All right. And so I get this the song and equal 0 to Infinity of -1 to the end -1 times negative one over in. Okay. Mhm. Oh I'm sorry. Good. We get not just negative one, we get negative one to the end. Right? Again subtle slow. I would have this if I did that carefully, which is why I get negative one to the end. All right, good, good catch there. Okay and now this kind of is starting to look like an alternating series and alternating harmonic series, but it's not okay because if I take a look and simplify the numerator, Get -1 to the two. End -1, adding my exponents. Okay And if we check that out, -1 to the two N -1. Well that is -1 to the to end Times -1 to the -1. -1 to the two. End is one Times -1. So that is always -1. Right? And so it turns out it's not an alternating series, it is just -1 over N. Which diverges. That is the it's the opposite of the harmonic series if you guys want to see it like this also. All right, that is the harmonic series that diverges so I cannot include that endpoint. Okay. All right, that was a little little tricky there. But um if we're slow and steady we can get it. Right, okay let's check our other end point here, Let's check out what happens when X is five. Mhm. So again I'm just gonna put in five for X in my series. So an equal 0 to infinity -1 to the end -1 Over N. Times 5 to the end. And now it's going to be a 5 to the end here. Okay, for X. And now this time those five to the ends completely cancel out. And I am left with Some and equal zero of infinity -1 to the end -1 over in. And this is an alternating series that goes to zero. The terms go to zero. So that converges by the alternating series test. Yeah. Okay. Okay, so that one was a lot simpler again, Five as an endpoint works, negative five doesn't All right, so my interval of convergence again it's going to be negative five less than X Less than or equal to five because I can Including five because that series does converge. And again, if we want to use interval notation, it's going to be open parentheses negative five 2 5. Close parentheses. All right. Super. Hope you guys are enjoying these have a good rest of your assignment.


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