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ToatCoocU Ocar( comteekcut O dietecniamim Rferine whclhxr Ilte: sries6Jipoints)Determine #hcllier the scrics S( the seties If Ic cuntereeng(o cunstant number!4contc...

Question

ToatCoocU Ocar( comteekcut O dietecniamim Rferine whclhxr Ilte: sries6Jipoints)Determine #hcllier the scrics S( the seties If Ic cuntereeng(o cunstant number!4contcirengdierent Find the sum 0f273.(5 points) Find the radius and interva convergence ofthe ~rics _04. (5 points) Find lim]275. (5 points)E valuate if possible X+[2J6. (5 points) = Find the Maclaurin series for f (r) = 2-=2() _

Toat CoocU Ocar( comteekcut O dietecnia mim Rferine whclhxr Ilte: sries 6Ji points)Determine #hcllier the scrics S( the seties If Ic cuntereeng(o cunstant number! 4contcireng dierent Find the sum 0f 273.(5 points) Find the radius and interva convergence ofthe ~rics _ 04. (5 points) Find lim] 275. (5 points)E valuate if possible X+[ 2J6. (5 points) = Find the Maclaurin series for f (r) = 2-= 2() _



Answers

Find the Taylor series for $ f(x) $ centered at the given value of $ a. $ [Assume that $ f $ has a power series expansion. Do not show that $ R_n (x) \to 0.$] Also find the associated radius of convergence.

$ f(x) = x^5 + 2x^3 + x, $ $a = 2 $

Okay, so we're asked to fund medical serious for half and its release of convergence. So what is Thie McLaren's serious flap necks at axe? Because two zero at his fines. So this is just the constrictive at Max as euro over in pictorial and X minus zero to the power of end and of horses from zero to infinity. Okay, so which part in this for murder? And we shall fund that. This is on President Victoria over in Victoria. So this is Gilles. It gives us an plus one times X to power if in. And what is this? So this is a derivative of that's the power of plus one, the extra power from this one. The X and equals with Justo Extract out. This the fish is operator. So this is extra power plus one and his friend zeros for infinity and equals to the derivative of Earth. Next comes this part's actually power of n and A from zero to infinity. And this is our familiar. So it was two x or one minus x o. So our final answer is just to find the virility of this part. And this is just equal to woman's as to the power too womans at minus once, one times x. So it becomes too. One hour square of woman is X and okay fund the readers of Convergence. The rarest convergence is going to be the absolute events. The absolute bell of axe is less than one, so that is the greediest of convergence.

Corn series representation of each of the five x. It's continuous derivatives. We have prime of X. is equal 5-5 expert chain role. And if we continue on like this here we're going to get 10 okay 25 to the five x. 25 each of the five x. And so on. And so when we plug in zero we got one than five 25 Than 1 25 and so on. Okay, so then when we plug that in we get one plus five X Plus 25 x squared over two factorial Plus 1 25, execute over three factorial here and so then uh huh that's what we'll continue being here. So this is just five squared five to third and so on. So our summation is going to be from five in fact to the end next to the end over and factorial and we have that from an equal zero to infinity. Now I need to find the radius of convergence. Let's take the ratio test into account here. Yeah. Five to the end here plus one next to the N plus one. Alright plus one factorial By that by in Victoria or 5 to the end next to the end. So that's going to be equal to five x over and plus one let me simplify that. The limit as N approaches infinity. That's going to go to zero. Just definitely less than one. Smart radius of convergence. It's going to be equal to infinity

So we have this function and we need to figure out me to find what is the register convergence around the point zero eso. Let's follow the theorem in the book. Um, 11.7, which says So you create if we have ah, functioning, therefore, lakes, which could be region, is it fraction? So, Pete, off X uh, functioning. Dino Meritor and Cure X ganeshan ended denominator. Um, the route eso routes off the function cure ICS from from these boots, we can't find what it during ready a so called charges. Uh, so Jenna in general, So these routes like points where dysfunction become zero. It might be on the conflicts point, so it might be complex number in your now s. So that's why I wrote Z Z i z za I indicates that we like several of them. So solutions 77 DC quit several. Oops. And so the way we can find the release of convert agencies, you just need to find thes times on the complex plane. Between these these points and the point. Yeah, we're considering you don't Beach were considering the expansion. So and we need to choose medium of distance. So minimum of these numbers minimal from these numbers. So this will be the radius of convergence. So let's see what is, uh, what have these numbers in this case? So what are these roots and what is the radius of convergence in this case? So Eurex, with his cure X in this case, cities X squared plus two x my strength. And so when does this become zero? So this is quadratic equation? Well, no. So groups might be S o Z. He's So you have minus here. You see the class? Oh, so you have two plus minus, uh, square root. Uh, these square. Yeah. Two squared, which is four minus for, uh, times the first car vision, which is well. And the third group should receive just 54 times by its, um 20 and over to and the first Gulf shendry his wife. So this is a solution on these are there, so that simply finalise. So we will have one. So to over two is one and plus minus. Ah, and under the square with minus 16 on, um, squares of 16 is four. So we can, um, could good for in front of the square. root and under the square root we will hear minus one. You just so we will have for for I. But we also need to do this too. So instead of for we will have to to I So So we have these. So z one year to road speakers. This is quite erratic. Uh ah. Equation. We had re expected toe cool boots. Zero is one plus two. I, uh, two is gone minus two. I Okay, so let's consider the complex plane. So here we have imaginary cards we have called. We're really part on the X and images on the white. So let's put at these points on this plane. So we have x. Not here. Zero. And so then we have, uh, so will thes thes numbers. So the real part and imaginary part real part send images. But so for the 1st 1 of the real part, he's one. So me, it's a one here in imager artist, too. So to re summer here. So the point will be a day interception so that it's real part. It's X projection is X x projection is one and imaginary. But McKinney's too. This is to this is one and this is you want no sooner early uh, zip for Z two z to ease. Oh, a real part off Z 21 I also, if we extend this lying so it will will meet Zito, sir. And so no dead's Zito eyes symmetry They're on the excites Servizi to everyone are symmetry Semitic from the sexes from the realism. So these will be minus 2 40 to imagine that these miners took well. So what are these decent since sees, So is you can see so these differences so they are equal. So it is enough to outweigh the only one one of these distances And this will be the answer. This will be the radius of converges as a well look at the right triangle. So these right tango here, uh, where so one side is one in other side is too. And, uh, we need to find the hypothesis. So this would be squared two squared that small square beaches where the back eso hour radius of convergence will be square to fight

Fun power, serious reputation for function F ax and determine the readers of convergence. Okay, so I'm gonna transport axe into lawn with this pool of five, and it's because against one ministers or five And what this This is long fine, plus lawn one minus X or if I and we're going to spend this part, actually, what is that? So, uh, this is just the integral of when there were one minute six or five the X. So let's have not. Maybe we should modify this by some constant K here, so let's try this. So, what is the Negro Levitt? Is this on long one minus plus to power? That's over five. And I think there is some constant here, so Okay, let's do this. So, what is the relative of long one minus? That's or fight. So this is one hour four, five times, not people one thing. Okay, so we confirmed that the way it goes, Nothing. One fifth. So, uh, yeah, okay. And actually, we could spend this part. So one of our women, this X for fighting is very easy to be expanded into hair Siri's. So this particle becomes to end from zero could be and, uh, extra power in your health. And here I have the differential of reader, the X. So we're gonna integrate this one through the term and actually were able to change the other integral. And the summation in there on his son. So on with the first girl extra card and ORF of love in the X and this summit. And this is just equal to our own life. By my eyes, when you were to the power from past one and some notation Inigo is explored in the eggs. It's rather familiar to us. So on. Okay, let's first you write this part and this is just lawn five minus when there are servants, US one and the integral is gonna be extra power and plus one over and us one. So that's the girl. Yes. Okay, so now we have our results. That's the power of serious reputation for the function for FX on determine the readers of Convergence. So how to find the richest convergence? We gotto push up on it from the step where we just expand dysfunction. So it's this equation we require that that's or fine is from zero ice from the ones who want question, place, axis from knocking fight to five. So that's the rays of convergence, okay?


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