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0/2 points Previous Answers LarCalcET6 9.9.006.Find power series for the function, centered at €c = -9rt2ix)Determine the interva of convergence (Enter your a...

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0/2 points Previous Answers LarCalcET6 9.9.006.Find power series for the function, centered at €c = -9rt2ix)Determine the interva of convergence (Enter your answer using interva notation:}Need Help?Read ItTalk to Tutor

0/2 points Previous Answers LarCalcET6 9.9.006. Find power series for the function, centered at € c = -9 rt2 ix) Determine the interva of convergence (Enter your answer using interva notation:} Need Help? Read It Talk to Tutor



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Find a power series representation for the function and detemine the interval of convergence. $$f(x)=\frac{x}{9+x^{2}}$$

In here. We have you on the function f x ego Jew actually running when I plus X Square here, let's try to modify dysfunction affects a later. I want to make the wannabe in this nine here so I would divide everything by nine. So have the X over nine divided by one plus now for the plus, I can reread it there as a miner's on minus x square. The reason why I converted this phone because record that I have ever have 1/1 minus X. The power that Siri's representative for this function here in is the geometric series here from zero to infinity and it's really fund absolute X more than one. So based on this formula, here we get exactly almost the same thing. So we need to do is would replace X hair by the minus x square this exhaled by the Manus X square, this ex hand by minus X square And then we should get now one of ah, one minus minus X squared. And now it will equal to the submission minus X square and from Serge Infinity And then we simply fight. And it was getting a coach of minus one about and expected to and from search infinity and it will be invited for the months Next square smaller than one. Our meal stop the afternoon X will be smaller than one and now it's not finished yet because we have to modify with this ex oven on the top Here. Therefore, on we need to do is removed everything here by the X number nine. Therefore, I have FX ICO, Jew X number nine Breast X square and we go Joe X over nine times the submission of the minus one about and expert you and from Serge Infinity are we can signify this one by submission from Serge Infinity minus one bar. I am expel Jew and less won over nine and it's funded for the absolute of X smaller than one. So we get the answer here.

The discussion will call that it will have the 1/1 minus X. If we go to the submission, the expel and from surge Infinity and it's followed found the absolute of X Smaller than one in this question were given the function F X. They called you the X, dividing by the nine plus X square. Now, the first time we need to make the night here becomes one and that you do that we will divide everything By nine, they would get equal to the X over nine, divided by one. Plus, we can wrap this time in June the thanks over three square all we can turn this into the miners and then this one will be the miners here like this. The reason why we do Uh ah, it should be. Be careful of medicine. So we have the minus there and then we have the minus off the X over three square symbol like this on now, the reason we do that because now we're going to replace angst in the formula here by this quantity here by the Manus X number three square. And if we do so, we should can, uh, 1/1 plus x square on the night. It will equal with you the submission from zero to infinity. They will have a minus X under three square and then we have the, uh uh sorry. This one will be the minus outside. Yeah, And then we have the about and and then we simply find this one again. The minus one Power n and then x about you and divided by three about you and and it from the zero to infinity. And it's valid for the absolutely minus X squared over night. Smaller than one are we get this one will be the absolute of the X square will be smaller than than nine on the absolute of X will be strictly smaller than three And then the last time we need to do will be We're going to obtain the function f X by Multiply everything here I just wanted to hear So I multiplied by the X over nine. So I have the function f X echo Jew, the X over nine plus x square and we equal to the submission from the Jew infinity. They have a minus one bow and and then we have the Expo two and plus one dividing by the three vow to and plus Nice. Now it will go to the three square. So blessed you here, Andi. Then it will be valid from the absolute of ex model and three.

Little thing to the last solution to the problem. Right. Uh So we did a solution to the last problem before this one. We're gonna do a similar thing whenever you have those ffx. Let me read it again. Whenever you have this function F effects to be uh you know, we want to right the infinite series or a power series representation of this function. Ah Well, uh so I'm gonna, you know, factor this is the same as this, first of all. It's the same thing because edition is communicative, I'm going to factor the 16 out. So when I factor that out, what do I have? Uh have um you know, uh X squared over 16 and I just have one plus exit a 4/16 have that. So once I have this one, um you know, what is happening? I have um I can rewrite this one as X squared over 16. 10 times 1/1 plus extra 4/16. Right? So this can be written in in terms of geometric series. So this is gonna be X over X squared over 16 submission. And from one to infinity um you know, exhibit 4/16 ultimate power and right, that is the geometric series representation. Uh In fact, I must write this one as a negative negative, right? Because that is how we're supposed to make it look. So this is what I have. This negative is gonna be here. Right? The addition one plus that is the same as one minus minus that. Right? So this is the geometric series representation and that is the same as the power series. Right? So this is a power series representation of the function. I can actually make it look a little bit more clean by taking this quantity inside here to merge with this one. And I'm gonna do that in just a bit. So I just want to pull out this negative one. To the power in right? As a negative here and has a power in. So I pull it up. Then what I have is this X squared, multiplied by this X to the fourth. It's just um mm hmm. Ah, excellent power for 18 plus two because this is eggs to the power four in and this is X squared. So when you multiply by the by the loss of disease, it's just gonna be forward to the power. Excellent power for M plus two. And then you have those 16 to the power of one. And that is multiplying uh 16 power in right here. So that's just gonna be, you know, 16 uh to the power and plus one. Right? So this is where you have uh as the, you know, infinite the paris series. Hey, this is the power series representation of the function given. Now we want to find the radius and interval of convergence. So we just you know that the geometric series converges like we did in the fast tutorial. It converges when x absolute value of X is less than one. If you do not understand this, please watch the last century before this one. Uh It is uh explain a little bit there. So we had this one. So what is going to be our X in this case? Is this one this guy right here. Right, that's right. Uh This quantity right here is gonna be our X. So in order for this to converge, we're going to require that the absolute value of extra power for over 16, it's less than one requiring this, we're gonna require this one in order to ensure that this one is convergent. So now this is just the same as extra power for 1/16. Less than one. Right? So we're just gonna have, you know, Excellent power four. Okay, less than 16. Right. I'm just taking this 16 to this. I just multiply through by 16 to get rid of this fraction right here. So that is the same as X absolute value of X to the power is less than two because 16 is the same as two to the power for. Right. So the exponents are the same so we can equate the basis. Right? So that is why I have this one. So if X to the pound four is less than 16, then X is less than two. That's that's the logic right here. So this is gonna be the interval of convergence, negative to negative to positive two. So this function this series here is gonna converge in this interval and the radius of conversions. That's too mhm.

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.


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