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(a) Find the interval of convergence and the radius of convergence for the seriesEo-1)"_ (21 + 3)"_ Vne +1Find power series representation about0 for the ...

Question

(a) Find the interval of convergence and the radius of convergence for the seriesEo-1)"_ (21 + 3)"_ Vne +1Find power series representation about0 for the functionf(c) cln(2 + 31)and specily the interval on which your representation is valid:

(a) Find the interval of convergence and the radius of convergence for the series Eo-1)"_ (21 + 3)"_ Vne +1 Find power series representation about 0 for the function f(c) cln(2 + 31) and specily the interval on which your representation is valid:



Answers

Find a power series representation for the function and determine the radius of convergence.
$$
f(x)=x^{2} \tan ^{-1}\left(x^{3}\right)
$$

Please. Yeah. Okay. So find the power serious reputation for the function and determine the raiders Convergence. So at that sequels X squared plus x times, the sea jets and Drea s equals one or women's X cube. And we got first expand GS and the expanding equals two from zero to infinity. That CDO the infamous extra power vent and the equations by the binomial theorem is going to be one plus three minus in class three, minus one or over and minus one. No, this over in. Yes. So this is going to be in from zero to infinity and plus two who is in Tom's asked to power in and we plug in Jack's into the equation one. So the fundraise off after Pat's he's going to be and, hey, signal from zero to infinity. And that's to choose in extra power and plus two class and from zero to infinity, the interest to over in absolute heart and plus one and the readers convergence is gonna be Are you close? One

Given from should f Thanks, Nico Jew X square plus X divided by one minus x about three record that it will have the 1/1 minus X when we called us so much and under X and goes from zero to infinity and it's valid from the absolute X more than one. So just this formula here, the first time I'm going to do is I would be riveting on both night. I'm just a question. Therefore, I should get the one on one minus x square. Now it would you go to the submission on the and, uh, expert and minus one from one infinite day and it's funded for the absolute X more than one it would do. A Nonda nearly would do on both signed now and initially getting Now I'm here as you go, June. Uh ah. Two times the one minutes expert off the a minus three times good. And the river the inside would be minus Kwan and it will ico Jude uh, ju minus 2/1 minus x about on the three. Now it would you go, Judah submission off the end temps and minus one expert and minus two and we start from to installing the one now absolutely exploded and one it was signifying. Uh and then with doing it to Mac this ex Cuevas X. Obviously. And we want to remove the minus X hair therefore and went to winter black Everything here by first I will remove The man is 1/2 so I don't mind multiplied by minus half here and I want to medicine appear so I will move to blind by the ex. Grab this ax now so they're far even do that. I will get this one examining coaches f X and on the right hand side here again the minus 1/2 ex grabbed us Thanks times Goodness Submission under and times and minus one X and minister from Children Infinity. So I should get Nico Jew and we give them minus 1/2 outside inside and we'll have us a mission for the X choir here I will have from judge infinity and and square minus and times with the X power And now and I would give them for bracket here, plus the submission from Children finishing now and square money span. And now for this X Now it will give us the expert and minus one now. And here in a chicken, by the job to one we need to make student were Mac person becomes a expel And now you need to do that. We can rewrite this one as the submission off We will stand from one instant. I'm from Ah June Now and then we should ghenda and bless one square minus and plus one and then times with the expo. And now here we have a submission under end square minus an expel. And then we have Ah minus 1/2 in front. Hey, we should get ism staff from two. And now we will get minutes 1/2 in front isn't because I'm from two. So you have this one will be This is a mission. And here everyone to stand from two here the first time It will be a need to press the first time here. So an equal to one and she getting code You Ah, one to square will be far Far Minister will be Jew. So blessed Ah, your X square now and then from here I'm should be able to do combined them into one single summation now from Judge Infinity. And then I have the expert and now in san the and scram Honest and And then I would have hand will be, uh, Plus for this one. I should ghenda and square Blessed do you and bless one minus and minus one. And then I will breast that Teoh X squared outside here. And I used to have a month and 1/2 outside. And so I have a submission from to infinity and here I have expelled. I am Yeah, again I can see that making consoled Ah ah this when we just wanna have a Jew and square and cancel out with this one with one and one concerned with this one So we have only the Jew and square. So I were edition as a Jew and square expert and less Teoh X square. Now notice that this to him we can put into the first into once your ah, submission Because when any could you one we get exactly this term here, So we have now will be minus 1/2 submission off the Jew and Square X and from judging from Guangzhou infinity now and to Higginson a loose two of them. And I'm gonna minus solution from once in 50 and Square Expo and and that's when we the answer.

Okay, So fun. The power Siri's for death backs and determine its readers of convergence. All right, so we can first, if spend these attendants of x Q part. So this is equal to and from zero to infinity and X cube to the power of to most wanna words who minus one times nothing wanted power from minus What? And we know that we knew it from the Taylor Siri's of octane in X. The riches of convergence is just articles one. And so this is gonna be seen with five to and from one suing for the X to the power of six and miners, three plus two is minus one over two months. One times ninety one off on minus one. All right.

Okay. Find the power. Serious limitations function and turning the readers of murders affects. It was That's cute. Comes one or two months X cure so we can use the trick. And we claimed that it was to execute times little one half times. The second one there were two minus X. So we're gonna fund it is true or not. So what is it? Good one? Or ask Teo to overtax whenever two mothers? Yes. So the first derivative of it becomes too when there are two months asked our two and not one. So it becomes too minus one or two months esque square and we can do this. Do it again. So the double direct vehicles the directive of this dysfunction it becomes, too. That derivative of minus one times two months. Extra power from one is two. So this is just two months Extra hour, one three times minus two terms ninety one. No, this is on. So this transferred out isn't one is two. One is to co over two months. Ex cute. Yes. And here we have number one half so naked one hat and later to the cancer dot with Siri's in a quote. This is It is true. Okay, so we're gonna continue doing this. We can read this part in things get easier to do because it's very easy to expand when you were two months X. Okay, so this is just to the cube. Next, ask Cube asked as Cuba or two times one half and second review of one minus half of ax Silla. It becomes too. Not you excuse or four times the second directive Extra power of end to tip our inn in from zero to infinity. And this is, um, begin minus X cube. Oh, four and weakens first through the first directive, Piso and from zero is showing family. But which out? So the first time it is extra power of zero over to deposit. So it is won and won Has the director zero. So now and should be former once evenly. So it becomes too X to power on minus one and toms and ends here. Yeah, otherwise we're going to find zero, so it doesn't matter, because if we put well, who would know it matters. So the sea if you put a little syrup in front to seventy. So there's a there's a mistake because X to the power zero minus one is is thie inverse of acts. So it's not true. It's not bad form for the power. Siri's. Yes. So that's why we should modify this in that from one community on DH. Yeah, that's sure, because when zero this term is a constant and now so the identity is thiss part and community doing the differentiation, it becomes two months of skew go for and here, hands front to Because we're in this one. The first part is the constant the first term. So can you bring that there, Load him for this part? It becomes then times in minus one Excellent, too powerful and managed to and over to the heart one to the power n So this is our final answer for the representation of power. Siri's. So this is the final answer. Yeah. Okay, so that's us to find the readers of convergence. So here we have the absolute off X over two is less than one implies I am still available and is less than two sword. So which means thie reduce of convergence is our equals two. Okay,


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