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Find the power series representation of f (2) convergence of this power series.about the center 20 (1 + 3i2)2Find the radius of...

Question

Find the power series representation of f (2) convergence of this power series.about the center 20 (1 + 3i2)2Find the radius of

Find the power series representation of f (2) convergence of this power series. about the center 20 (1 + 3i2)2 Find the radius of



Answers

Determine the radius of convergence of the power series representation of the given function with center $x_{0}$. $$f(x)=\frac{x^{2}-1}{x+2}, \quad x_{0}=0$$

Okay, so this is the function, and we need to find the readies of corn emergence school. Uh, dysfunction around the point X node equals zero. If we consider an expansion out the point X not equals. Okay, So, well, uh, healthy serum in the book. Uh, Syrym Now I'm there 11.1 point seven, which, which says the following. So when? Well, so we write a function if we can write the function. Ah is as fraction a za fractions or x p x Correspondents toe the function in day numerator off the orginal functioning on cue exit to function on day don't nominate. So if we can do that, uh, then all right, we consider the function in the denominator. And if we consider routes off the function points when the function becomes seal So, um, then we can we can find what is the radius of convergence based on these routes? So ah, root, Who's generally could be complex number on the complex plane. So, um, that's why I wrote to see, uh, indicating the general conflicts number. And I indicates that we might have several of them. Well, we might have several roads in case of several routes. We need to Ah, In order to find the radius of convergence, we need to find the minimum distance from this point on the complex plane. Ah, and the point around which were considering the expansion so we can write the readies. A convergence will be minimum from these numbers distance between the point and, uh, the point around which were considering the expansion. So it's not, um Okay, so this is a serum. Um, Now, listen, this use this three room toe find what is the radius of converges in this case? So what is a Q? What is this function? Que in this case, O que of X. In this case he's expressed to Ah, what is the what is the root of this function? Uh, when renders function becomes zero at the points Ah, minus one. When x equals my when xy equals minus two, function become zero. So the route will be minus two. Uh, and as I mentioned, so we recall these routes z z one z two z three etcetera. Even though in this case, the the road he's rial In general, we might help. Ah, complex number here. Uh, there's wise ease or more really want really want? Uh, so we have one only one route because the function is linear. Ah, and so it is only one solution. This was expected and now we need to find these distance. So what is Ah Z one minus two? Eso distance between these two points. Well, let's do these graphically. So it's going to see Derrick on complex plane. So we have a real part. We have imaginary parts off. The numbers are in this put thes thes two points on the complex plane. So they are both are real, so it's not a zero. So you can put extent here at the origin, and Z one is minus two. It is also on the re. Alexis ser minus two is here and the distance between these two points, he's obviously so these distances to, uh, so readies of convergence is too

All right. So this is our function and were asked to find the radius of convergence, uh, for these function than we expend eat around at this 0.0. So Well, so we have We'll decide. See urine in the book. So you're in Well, you live in 0.17 which says that if you have a function which has fraction form so way have end of eggs, which is sound fraction function. Your mix, um, over. It's a you have the function Q of eggs. We can say that the readies of corner just depends on yeah, the routes off the function in the denominator. So you we right? Que z, Mom, let's a Does he know? No. Generally. So these routes might be complex numbers. So let that's why I wrote to see eso in desert, uh, roots off dysfunction. Meaning that if these points function becomes you, you know? Okay, um then the radio convergence will be let me write Meaning moon from no these times between these routes and the point around which were constituting the expansion. So Z does he know? My yes takes note. So instead of three not literally, right. See high so indicating that me white help We like hell down the number off under fruits, number of coins win these functions. Come, see. And we were choosing the meaning again. It's, um see what we have in this case. So our q q or X is x x squared, sworn And, uh So where does these fraction becomes becomes, You know, it becomes CEO when x Well, I a right to receive one. Uh, because it is complex. In this case, he's, uh I complex number because I squared is minus one and plus run to samba zero or we have the two, which is mine beside. Okay, so we have to complex roots and way. No, the point, they're only which were considering the expansion. Okay, so now let's, uh, put these points on the graph and see what happens. So what What is these thieves times and what is the radius of Conrad's? Okay, so we have this point x known. Yeah, zero so explode. And then we have these two points. So these two points are complex. They will be along the y y x is, uh so why I'm going to be here And these point correspondents. I these brain on chorus price minus side. Uh, and then you need to find the distance. Uh, so these these point is one. This is going to my distance. Eyes received is one. So, for for both cases, the distances one, So minimum is also one. So the radius of conversions, he's simply more.

Find the power serious representation of a function and determine the readers of burdens. So after facts, because one person likes times the derivative of one over one minus X in terms that one. So let's see if it's crept. So one hour was X and everything is gonna be when you work. One one is just my woman's X square and like one here. So you have filtered into one. So this is true, okay? And we continue. We can expand this part so it becomes minus one minus X guns, sir. Zero from any from zero to infinity. An expert in And is it just knight to not to go from zero to open day expelled in minus stew and from zero to in any hour end class work and the fun the reserve burdens it requires absolute. That is less than one. Which place are equals. What? Okay,

Function F x decode you one just x divided by one minus X out. You here. We conduct Ever have the one off? One minus X It would you could you know so much on the X and from Serge Infinity and Denise van it for the absolute ex miner than one. So now imagine getting about you. I would do that. The rear of the ambush. I am disc one. So if we do so and again Uh huh. Hey, I grew again, The 1/1 minus thanks. Ju under left hand side and on the right hand side under so much in the end times with the expert and minus one from such infinity. And it's welded for the absolute X smaller than one. And now we will have the one more stamp to do with the numerator. Yeah, so I wanted about everything here by, uh, and because I want us thanks. Times when this one. And then we should get, uh, the hunting here. We could, you know, have X Now and then I should get equal Jew, the Congress x times with that. So much an absurd infinity and times wouldn't, uh Thanks. Bella. off and minus one and then have absolute X monitor and one. And if it was getting now that so much time those urge infinity and expert and minus one and it isn't gonna be there from one Istana Mazo now and then we have the blessed we have the X, Then then we can wound up and inside. I know from one to infinity doesn't from one to infinity and absolute of X smaller than one. Yeah, we can redo this one by noticed that we can push everything to the barrel off the end here. So for the first case here and we write it down as the ah, there's an increasing isn't I wish them from zeroing sternum from one. Now, in this time, from zero is one becomes the and just one. Now this becomes X n plus submission from one to infinity and expel. And and now I will have the first time in this case will be equal to one. And then we're plus the submission from one to infinity and press one expert and plus submission from one to infinity and expert And and then it can ever be what you combat you have damage one. So have one plus submission off the I will have here will be the endless one bus and times with the expert. And now from one to infinity. And if I should get one Plus, this would mean that do and was one No X and from one to infinity so and this And in here we can combat you of magnitude. One single submission has starting from Suraj Infinity to and this one times expert and and it will be valued for the absolute of X more than one. So this will be the answer here.


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