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Work Problem 2 (45 points). (a) (10 points) Use the power series representation 1- Zizo x" to find a power series representation for the function f(x) 6+1 (6) ...

Question

Work Problem 2 (45 points). (a) (10 points) Use the power series representation 1- Zizo x" to find a power series representation for the function f(x) 6+1 (6) (20 points) Find the radius of convergence and interval of convergence of the series (-1)" Eaz] n'3" (c) (15 points) Find the Maclaurin series (that is, the Taylor series centered at a = 0) of the function f(x) ex

Work Problem 2 (45 points). (a) (10 points) Use the power series representation 1- Zizo x" to find a power series representation for the function f(x) 6+1 (6) (20 points) Find the radius of convergence and interval of convergence of the series (-1)" Eaz] n'3" (c) (15 points) Find the Maclaurin series (that is, the Taylor series centered at a = 0) of the function f(x) ex



Answers

$15-20$ Find a power series representation for the function and
determine the radius of convergence.
$$f(x)=\frac{x}{(1+4 x)^{2}}$$

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.

We want to find the power series representation of this function. And we're gonna use a we're gonna employees similar tactic do of what we used in the previous our tutorial. We're first gonna take the derivative because that is easier. It's gonna get us the geometric series and then we can use integration. So what is the derivative of this one? But this derivative is negative one of a, you know, five minus X. That is a derivative of this function. Uh Now I'm just gonna factor out those five, you know, from this denominator. So that is gonna be negative 1/5. What is going to be left is 1/1 minus X over five. So this this one is a geometric series, right? And so we can apply to formula. So this is gonna be in from zero to infinity of, you know, negative one to the power. And in fact, I think I'm gonna run out of find out of space. So I can just right here. So this is negative 1/5. And from zero to infinity of negative 13 Power in rectus. This it's actually positive uh X over five to the power in. So this is the derivative. Now we're gonna find the power serious representation of F. Not the power serious representation of F. Prime. So in order to get our X, we're just gonna integrate both sides. That's it. So whenever you integrate F private respect to X, you're gonna get therefore fax in this integral sign. Because this is continuous, you can just go through the summation and we can bring it here. So when I bring the integration there and I try to integrate, what I'm going to get is summation. And from zero to infinity. Uh and eggs to the power, you know, N plus one over five and times and plus one. And there is this is the uh the integral. Right? That is the integral. And then plus a. C. Right? But we can Mhm. We're gonna find a see you later. Ah So I want to bring this five here because you have to of the five, you have five to the power one and five to the power in. So when every day they multiply, you're going to have something like this. Excellent caller and plus one over 53 power in plus one in times and plus one like that. Then proceed. So this is kind of the uh power serious representation of effects. Now I want to find the sea, right? Uh So we want to find a C. You know this Apple face is natural log five minutes X ray. This one. So what is F. Zero F. Of zero is natural log five minus zero. Which is just natural level five. Right now, I want to read put zero here in this expression, you can see that uh the entire thing here is going to go to zero because if X is zero, this summation is going to be zero throughout zero plus zero plus zero. You know, everything's gonna be just gonna be zero. So we can now use it to find a C. So fo zero, which is natural log of five is the same as see, right, when I put F zero here, wherever I see X, I'm gonna put zero and it's gonna make the entire thing entire information here. Zero. So it's just gonna be left with C. But at 40 is natural log of five. So C is natural log of five. So finally F of X is going to be summation and from zero to infinity it's a negative year. Excellent power hand plus one. Five to the power and plus one the name plus one then uh plus uh natural log of five. So this is the power serious representation and I wouldn't find the radius of convergence. I can use the ratio test, right? So I can use the ratio test or I can just use the yeah, the first derivative, you know this one? Because we said in the we said that whenever a function converges uh in an interval, then the derivative and the integral also converges in the integral uh in the interval. Right? So uh the convergence of the derivative here, the radius of convergence of this derivative is the same as the radius of convergence of the function itself. And what is the radius of convergence of distributive? Remember this derivative here is this summation and from zero to infinity. Excellent power in over, you know, five to the power in plus one. Right. Uh that is the that is that is the power serious representation of the derivative. That is what we have here. We just integrated. Right? So this is the power serious representation. And we're saying that the radius of convergence of this one is the same as the radius of convergence of this, of the series this one. Right. So uh you know, you're gonna have eggs over five X and or five and you know, less than one. Okay. Yeah. And so you're gonna have eggs in absolute value, less than five. And so X absolute value is less than five city radius of convergence for the derivative F prime this one. Right? The radius of convergence of that is five. But we said that the radius of convergence of the derivative is the same as that for the original function. So the radius of convergence of ffx is also, you know, also amounts to uh five. Right, So this is the power serious representation of F prime this one. Okay. And the radius of convergence is just uh taking the absolute value of this one and making it less than one. That is for a geometric series. So that is what we have here. And then finally we had this as the radius radius of convergence and we're saying that that is the same as the radius of convergence for the ffx, which is this one? Somebody can I can actually use the ratio test. Okay. You can use the ratio test but we're just using a theorem right? In a series that whenever you have a function that is represented in a power series, the radius of convergence of this function is the same as the radius of convergence of its derivative, and it's the same as the radius of conversions of its integral. Okay, so that's just what we use. But you can also use the the the ratio test and it's gonna work just fine.

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,

So we have this nice looking function. I think this function is a very friendly function. Friendly function. Friendly friendly in quotes. Let me just take it off. So X squared over one minus X cube. Okay, so we want to work on this thing. We want to find a power serious representation of just this guy. Once we do that, we're just gonna multiply by this Moon reader. And then we're done. Also we already know this format. We really know this one. So you can see that this one looks like this one is just that. It has a cute. So in order to get this cute, I think you have to differentiate this one twice. Okay. And definitely the first time you're gonna have squared and differently. The second time you're gonna have twice. Okay, so whenever I differentiate this one twice and I'm just gonna have this one. Now, all that I have to do is differentiate this one twice. What is the derivative twice derivative of this one? It's just gonna be in from 2 to infinity. Uh and in -1. Hoekstra Power in -2. That is the second derivative. And that's it. God is the power of serious representation. Okay. Uh So we can go on and find the radius of conversions. Oh I beg your pardon, That is not a part of. So we supposed to multiply by this guy. So whenever we multiply by X squared plus X, then it's just most find this one by X plus X squared. So what we have is in the face equals uh Summation. And from 2 to Infinity this is going to be eggs to the power. Excellent power, naked and -1. Whenever I multiplied by this one, when I multiplied by this one, then it's gonna be plus summation. And from two to infinity An and -1 exit pile again. Now I'm going to combine the summations. So this is gonna be after ffx summation and from zero to infinity uh and multiplied by and minus one. Multiplied by, you know, Exit power in -1 And -1 plus exit power here. That is what I have. So this is the para serious representation. We can try to make it a lot nicer because whenever we want to find the radius of proof regions, we're gonna struggle with this one. So you can make it a little more compact. Let me see this thing, 1/1 -1. When you take two derivative of this one is going to be two or one of my inner space. Cute. So there's supposed to be a two here. So in order to get rid of that too, I'm supposed to divide To multiply here by one half, so one half, one half and that is going to cancel this too, so is going to affect things uh Here is gonna be one half, here is gonna be 1/2 Okay, okay, so uh you know, I still want to try, I'm still trying to find a way to make this thing more compact so that I can do the ratio test and I think I might just, I might have just found a way uh Okay, let me take this one off. Mhm. So you know this thing here is going to be this too, I can bring it here, so it's just gonna be uh summation that Okay, extra power in -1. Big problem then over to, I just brought those two here, I'm going to bring this to here as well, so it's going to be plus summation and from two to infinity and -1 uh exited power in over two. Again, what is somebody going to do If I put em plus one word I see in here, it's going to allow me another, ready to change this summation to start from one to infinity wherever I see. And I'm gonna put in plus one, so this is going to be in plus one. If I put em plus one here, you can see that, I'm just gonna have and if I put in plus one here, I'm going to have exit the power and and Justin then over to Okay, So you put into this one here and they change this 1-1 right? Uh, trying to make it a little more concise than plus. Now look at this one, okay, this guy here is the same as submission and starting from one to infinity of the same thing I'm gonna explain in a couple of minutes, a couple of seconds. Sorry, do you know why? Because whenever and is one If you put one here when N is one, this is gonna be one, This is going to be one And 1 -1 is zero. So it's going to make everything zero. So starting from one does not change this one, it's still the same thing because he started from 10. But I particularly wanted to start from one because I want to make it look like this one. What now? I can do some combinations. So when I combined you can see that I can pull out the submission. What else can I pull out? I can pull out an end because the end is common. I could I could pull out a 1/2 Because 1/2 is common. So I'm pulling out and then over to I can also actually put up pull out an X. To the power end. Because there is so common So that I have is n. Plus one Plus and -1. That's all I have left. I have this guy and this guy left so I just add them and you can see that this is going to take away that right So if that takes it away I just have twice of this. So that is a pretty nifty where you're putting it because it's gonna help you a ton with the ratio test. If you put if you left it in the way that I did in the first time you're gonna struggle to do the ratio test because it's actually gonna be crazy it's going to get out of hand. So this is gonna be Times 2 to the power to. And because this is this is taking away this and this is adding this one to give you two. And and actually this too is canceling this too. So it is getting way simpler by the minute and this and is multiplying this. And so finally you can see that in a concise manner. This thing is in squared X squared no X. And in a concise man. So I've been able to manage to reduce the series to this floor which actually works in my favor. So this is the power of serious representation. And this is why it works in my favorite. Look at a ratio test. The ratio test is looking for a influence one or eight N Less than one limit and approaches infinity. It is very easy to construct the sequence in plus one in the end. So what is a N. Plus one? It means I would probably see in here. I'm gonna put end plus one that is N. Plus one squared X. To the power and plus one that is my and plus one. Okay, so this is going to be Excellent power and plus one. And what is my hand? My hand is just the same thing because and I wrap some absolute value around it and put a limit and approaches infinity here and let it less than one. That's my ratio test. So you see how uh huh simpler it has become put in this uh series into this form. That's a lot of heavy lifting for you. Otherwise you're gonna stroke you're gonna you're just gonna struggle in that gigantic sequence so I can do something a little bit. What can I do? I can um This is going to cancel this of course. So you just have an X. Year. So this thing is limit and approaches infinity of n plus one squared over and squared. And how our next year less than one. Because this is this is going to be left out is the X. Uh logic do taste that I should uh you know expand this breath disease which I'm gonna do n square plus and squared plus two. N plus one over N squared X less than one limit. And approaches infinity. Known a divide both sides by N squared, divide every term this term that that and that when I divide each and every term of this uh fraction buy in square to have limits And approaches infinity of one plus to over end plus one over N squared Over one times my eggs less than one. So whenever I let uh you know and gold infinity this is gonna go to zero. It is going to go to zero. So I just have one and multiplied by X is just my ex, less than one. So my radius of convergence is just one. So that is the essence of making the series as concise as possible.


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