1. (25 points) Find the interval of convergence of the power serics...

Interval of Convergence Describe the interval of convergence of a power series.

Discussion. We're going about the function one other one minus X and has the threes under farm expert and from the infinity. And it's valid, Found absolute of EC strictly smaller than one. And now a good replaced the angst in this formula here by the far X square. Then we get 1/1 minus for X square. It we go to the submission on the four X square power and from search infinity. And we simply find this one agenda for about an expert you and from surge infinity and it's valid. Found the absolutely for X square smaller than one we get. This one will be the absolute of X square. It will be smaller than one off a far. Uh, the X will be absolutely X will be strictly smaller than one off to. And then here we want to modify. The last step will be we take the function F X equal to the 5/1 minus four x square, which you need to multiply everything here by the five. Then we get in what we call gender. Some motion off from the Jew infinity five times the far end expert you and and then it's very found. Absolutely x smaller than one half. Okay,

Overall coverage is for this. Let's use the ratio test to do that. So Next to the 5-plus 5. Do it plus one Play five guesses stop or m plus one factorial multiple rather. Super Cool. Which is in victoria. Well, very next to the Simon. Okay, so when we simplify this here, right? The N factorial cancels with this leaves us with N plus one. The extra five ends cancel here leaves us with X to the 5th. So when simplified the effects to the 5th over and plus one at the limit as N approaches infinity, this Chico zero just always less than one, which means the interval conversions. It's from negative infinity to infinity.

It's going to require a bit of fairness. Eight one plus X over one minus X squared. This is gonna require a bit of tactic. You have to be tactful but that is the importance of this tutorial to help you practice and to help you you know blood you through some tactics that you can use to solve certain problems. I'm just gonna split this one into two. Right? This is applicable and exploding it into just two things. Right now I want to find a partner series representation of this one. This one looks exactly or a little bit like this geometric series thing. We know that this already is given, you know this one from antiquity that whenever you have something like this this is that right? But this time it is squared. So we're gonna here is where we're going to apply uh some finesse. Mhm. Okay. I know this one but I have this. So how do I um Mhm. How do I, you know? Yeah do this. Mhm. Uh In order to give this one from here, I can differentiate this one. Right? So whenever I take the derivative of this thing, D. D. X. 11 minus X. Because DDX of the Sky, I know that the direct about this one is going to give me this. So what is the derivative? The derivative of this 1? You know this function? This rational function can written last one minus X. To the power negative one. So when I'm writing the derivative, I'm gonna use the chain rule. I'm just gonna bring this one down. So it's going to be negative one On -1. Take away one from here is gonna be negative to multiply by the derivative of the inside. What is the root of this one? The inside is -1. Right? So negative one takes away this negative one. So I just have one over 1/1 -1 squared. So I have that medicine derivative. And you can see exactly that. I have uh I have this right here. Right. And so I got to deal with the root about this one. Remember I want to write, write a few terms here. If you write down a few terms, you can see that uh you know one and zero. This is one because it's going to be extra power zero. And that is one. If you take the derivative of one, it is just zero. Okay, so that is going to make this end start from one because the zero is wasted by the derivative is wasted. So you have this one. Okay, mm -1. multiplied by an right? That is the derivative of this X. So the power series representation of this guy is this guy. But in order to get this to make this app affects have to get this thing. So I'm just going to add that. Uh So to both sides. So this is just gonna be ffx. It was 11 minus X squared plus X. One minus x squared. Okay. Uh In fact, I didn't even have to uh you know, I didn't even have to split it. I mean this this one is going to work fine. All I gotta do is I believe multiply. Let me see to make it easier. You know, this is that right? In order to get ffx, I just I just have to multiply this By one. Close x. So when I do that didn't have Yes. Hey, which is very simple. It makes it a little bit simple. And so this is gonna be one plus X dense summation. And from one to infinity and eggs, I mean put it in this way, it's also fine. I just wanted wanted to use this one. This is a little more compact. Okay, so this is the powers of his representation. If you will somebody can decide to foil. That is not a problem, I can foil. So this is gonna be summation and from one to infinity X. And to the power X minus one then. Plus now whenever X kids here it's gonna take away this negative one because this is the power one. Right? So when I do that, what I have is summation and from one to infinity into the power X. N. So that is the power serious representation. Mhm. Actually I can do something more to make it more concise. I can bring out a summation since it is common. And then I'm just gonna bring out, I'm just gonna have exited power and minus one plus exited power in then this is common as well, so I can uh make it bring it here. Okay, so this is a concise for much nicer presentation, this is the power serious representation of F of X now onto the radius of convergence. Okay. Uh Okay, so mm I'm gonna use the ratio test again, your ratio tests again. Uh So I'm just gonna do it straight away because I've been explaining this one A lot of times it is gonna be n plus one exited power and Plus exited power and plus one the number An extra power in -1 plus extra power in. Yeah, I want to take the limits and approaches infinity, right separated a little bit And I'm gonna make it less than the one. Okay, so uh you know, this is gonna be a little messy. Uh huh. I want to know. Yeah. Yeah. Mhm. Yeah. Yeah This one here is a little messy. So what I can do is try to make a series better looking. I'm going to make it look better sir I can perform this ratio test. I mean I can do this one but it's gonna it's gonna drag and drag on for for a while and we're trying to make this video as concise as possible. So uh let's let's do this this let's rephrase the series. If you write out a few terms what do you have? Let's write out a few terms. Um You know whatever X. Whatever end is one what we have we have uh you know put in to be one here. This one is gonna be one this one is going to be one because this one So extra zeros one. So basically what I have is one plus X. one. And is to put two here put two here, two. Here you're going to have uh you're gonna have two X. Plus uh two X. Squared, right? One N. S. Three. You have in three X. Squared plus three X. Q. And blah blah blah. So what is happening is one plus this X. Is gonna is gonna add to those uh two X. So it's just gonna be three X. This two X squared is going to add to this three X. Squares is gonna be plus five X. Squared. You can see a pattern, right? You can see that this three X. Squared. It's going to add to a certain four x. squared. So she is going to be plus seven X. Cute. Right? This is supposed to be cute. Sorry, So you can see a pattern. So this is gonna be nine X. 24 and 11 and blah blah blah blah blah. So can you collapse this one into a submission for let's start to collapse it into a submission for so this app effects is going to be summation and from one to infinity something is coming through, write something, something is common. You can see I can see annex in because this is exited power zero exited power one, exited power to. So there's definitely uh there's definitely an ex N. Here that is risen with respect to the to the end. Right? Because here you can see X. To the power of your extra power one. Extra power to extra power three. So you have an accent. Now what else do you see? I can see uh an increase in odd numbers. This is one, this is three, this is five. So it is an increase in odd numbers. And how do you write odd numbers? It is just uh two and plus one. You write even numbers two and and write our numbers two. N. Plus one or two and minus one in which way? So see this one is a little more concise of a series than this one. So hopefully let's see if that is gonna make me help me do this ratio. Test easier a little bit easier to make the video as concise as possible. So I want to do this And make it less than one. So this is gonna be limits wow whatever I see in here and putting in place once it's going to be and plus two. Excellent and plus one this is gonna be over two N plus one. Of course it makes it a little easier. So this one is easily is it easy to deal with? So this translations is gonna go on. So I'm gonna have limit and approaches infinity. I'm just gonna have an ex left here and this is gonna be to end plus 2/2 and and plus one Then times this X is less than one regus. This is going to cancel this and just have one X left. That is this excuse scene. So uh this is what I have now, I'm gonna divide each term of this fraction by an when I do that I'm going to have limit and approaches infinity. Uh you know, to close to ruin over to close run over and then multiplied by eggs. Last time when I let and go to infinity can say I'm gonna have to over two which is just one. So I have this one. So you can see that the really is a convergence are it's just gonna be one. And then this is a power serious representation. I did I put it in this way because that is going to help me do the radius of convergence faster. But I mean, uh this is the parachutes representation. You can put it in this way, I can put it in this way. I mean, it doesn't matter.

In order for this infinite Siri's to converge, we first must make sure that the limit as n approaches infinity of this expression zero So we can set up the numerator so that the expression will be less than one. The absolute value of this expression will be less than what And by using the second tomb, we can arrange it so that the new movie will evaluate to be fight the power of zero, which will be less than or equal to one. I have said the power of negative end can be rewritten as 1/5 to the power of end and sitting X minus three to be 1/5 will give us the bounds. So evaluating for X minus two equals 1/5 will give you 16 5th and for X minus three equals negative. 1/5 you will get 14 5th. So we can say that the bounds The interval of convergence for this if it's Siri's is equal to two 14, 5th to 16. 5th includes