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Use multiplication or division of power series to find the first three nonzero terms In the Maclaurin series for the function: (Enter your answers as comma-separate...

Question

Use multiplication or division of power series to find the first three nonzero terms In the Maclaurin series for the function: (Enter your answers as comma-separated Mist ) Y = 3 sec(4x)

Use multiplication or division of power series to find the first three nonzero terms In the Maclaurin series for the function: (Enter your answers as comma-separated Mist ) Y = 3 sec(4x)



Answers

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function.

$$y=e^{x} \ln (1+x)$$

The problem is use much application of division of power. Siri's to find the first of three Nancy returns in the micro miniseries for each action. Why is he going to obtain in the X Squad? So first behalf I can Didn't axe? Is that true? Linus X Cube over three. Last access Over Linus back. Someone selling Don't don't don't anything cams X minus x Q three x two over far minus x seven over. Someone starts dot dot But the first German MRIs X How's Max exploiter? Generous axe terms Negative facts cubed over three This town negative X cubed over three last negative X cube over three times Specs, then thirty. Germany's Axe Holmes, thanks to the power five over minus plus Tax Cube over three times X Cube over three. Plus I just power foul off our backs Start. This is equal to a tax lawyer minus to food. From sex to the problem for you. Class is this is twenty three over forty five. How kind towards Apartment six last night

Now we want to do, uh, why equals x four sine x once again, if you've not watched the previous tutorial, I strongly urge you to watch it the last each other before this one. Because this is gonna follow the same thing and is usually gonna make you understand better if you have knowledge off the previous the nasty total before this one. So what is this MacLaurin series for this? It is summation. And from zero to infinity, negative one to the end. Exit to enclose 1/2 n plus one factorial. If we try to expand it, this is gonna be, uh, is gonna be X minus X Q over six factorial Okay with three factory, which is six. I'm just reading a straight away. Right. And then ah, plus exit of 5/1 20 disease five factorial right on an exodus seven over. So this is gonna be seven. Factory 1 to 7 factorial it is 50 40 right? That is seven factorial 50 40 eso you can go on and on and on and on, right? Eso we're gonna divide Just gonna use the 1st 3 terms that did so And we're gonna divide that this one, which is thes guys here gonna divided by this. This X right. So the X is the numerator. So that is what? What is going to come here? Good. Now in eggs, Device X. Just have one on one times. Everything is gonna be, uh, eggs minus skway X cubed over six. Right, Uh, plus exit of five or 1 20 And the use of tract basically, you subtract. Uh, this one is gonna be X. Ah. Cube over six and miners exit. If I over 1 20 then he's devised that as well. Then that becomes, uh, positive. X squared over six. Right. Exercise this one. That is what's happening. Uh, and then you multiply that by everything again. So this is gonna be X cube over six, then minus X to the five exit of five. Over, uh, 36. Right. And then this one is gonna be Exodus seven. Surplus, Exodus seven over, Uh, six times. 1 20 What is six times 1 20 Ah, 7 20 Okay, so you're gonna be 7 20 Good. And then, uh, so let's perform one more division. Right. Perform one more division. So this one is gonna go away cause dis minus. This is canceling out. Now I'm gonna have a negative x five, then minus minus. Serviceability plus exit. If I were 30 cities. So you just have to find the LCD and then make the, ah, simple addition of the fraction. So once you do that, you're gonna get Theo CDs. Um, 3 60 actually. So you're gonna get seven. Exit of 5/3 68 and this one is gonna be negative. Extra 7/7. 20. So that is what is happening. Right? So, uh, then you can have one more division final division, just doing three. So one more division and then we stopped. So X is gonna divide this one. The 1st 1 is there's gonna be seventh exited four. So plus seven x 2 to 4/3. 60 right. Seven Exeter four with 3 60 So, once you have that, uh, you can also continue the same thing. So, uh, we're gonna take this numerator here as the remainder, right? Going on and on and on with no problem. So this is gonna be one plus, um, x squared x x squared X squared over six and then Ah, seven x to the four. So seven extra 4/3. 60. We have this one eso That is what we have and that is approximately, um that is approximately eggs over sine X.

We can use the first several times of the taylor expansions to substitute E. To the power or N. Si and Sina Harris Becomes one Plus X. Press X squared over two as X cubed over three. That's a higher order terms of X cubed and times Eriks miners x cubed over six. Uh huh. Higher order terms of X cubed and then squared. Yes because we just coffee the first factor and then we can multiply out second sector Which is X. spared my nurse extra force over three. Trust the higher order terms of ERic's true or false. And then we can multiply these two factors out and then class higher order. Oh eric to afford so does he calls Eriks squared plus ERic cubed crossed 16 extra force plus the higher order terms of extra force. And this is just the first three term of this function.

So I say we have this one. And, um So what is What is, uh, Second X is the same as one over who signed X ray. So once we have that, uh, what is the MacLaurin series for Consent X. That is summation. And from zero to infinity, negative one to the end. Extra two in over to end all factorial. Right. So this is it. And I mean, we can we can try. Teoh expanded a little bit, So this is gonna be one minus, uh, X squared over two factorial and then close this time is gonna be exit of four for four Factory. The next one is gonna be Exodus 6/6 factorial then is going on and on and on and on and then But we have to, uh, divided have to divide one by this expression. Right? So if you divide one by this expression, uh, that means that we have to put the Siri's under this one here. Right. So, uh, this turn it express like the manufacturer reels to view numbers. Right? So this two factor here is for So this is for this is I believe, 24 right? For factor is training for and then six Factorial is 7 20 And then No, no, no. So now we have Teoh do pretty much a long division because we're dividing this one by one. Right? So let's just use the 1st 3 terms to you do that Do are bidden. So this is gonna be one minus X squared before plus X to the 4/24. But I will divide in this one. So once we do that, that division this one is going to divide this one once they multiply is gonna be one minus X if x squared for this X for over 24 and then you subtract. So when you subtract, then you're gonna have, um, as it four supposed to be to Yeah, here's to two factory. Here's too right. Do that. Your is to not four eso to here and then to hear, I believe yeah, in two years. Good says to you and to So when we subtract now, this one is subtracted, so it's gonna be zero. This one has new, uh, current a part of something. So we're gonna just, uh, using a Subtract it by subject it from zero. So this is gonna be is gonna be zero minus minus right? X squared over two. So just gonna be positive after all. So it is gonna be X squared over two positive, the same baby gonna subtract the 3rd 1 from zero, and it's just gonna be negative. Right? So this is gonna be negative exit for over two times 24 right? So once you have that, now one is gonna sub one is gonna divide this one as well. There's gonna be positive. X squared over two. Red one is dividing this guy right here. And in those times, that is gonna be, uh, x squared over two. And this times that is gonna be minus X 2 to 4 over four. And then this time is that is gonna be plus extra six over 48 24 times to his 48. Right now we do another, uh, such subtraction. Right? So this attracted by this is gonna be zero, and then this is gonna be, uh, this is gonna be plus is gonna be negative exit of four or between four. Right? Plus, because it's minus minus. Right extra to 4/4 right. So you need to find the, uh, LCD, You know that? What is what is the LCD, uh, the LCD off? That is 24 right. And so when you do that subtraction, you're gonna get five. Let me just bring this to to make a difference. So it's gonna be five, because the LCDs to enforce right. So just add this fraction, you're gonna get five exit of 4/24 right? And then this one has no kind of heart. So it's just gonna be subtracted from zero, so it's gonna be negative. Uh, Exodus 6/48 right? Anything of this raft work here? So we continue, and then ah, one is gonna divide this one as well, right? So when one device, this very one here and one devised, this one is gonna be the same thing, right? So that is gonna be also at a top five. So five exited for over 24. It says that in every here and then this is gonna some multiply everything and then bring here to the bottom here and then subtract. Right. So that is gonna be


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