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R+2+3x+29) Find the following series and their radius and interval of convergence: a) MacLaurin f (x) = 2" b) Taylor at a = 2 , f(x)=x 2x'+x c) MacLaurin ...

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R+2+3x+29) Find the following series and their radius and interval of convergence: a) MacLaurin f (x) = 2" b) Taylor at a = 2 , f(x)=x 2x'+x c) MacLaurin f(x) = d) MacLaurin f (x) = In(

r+2 +3x+2 9) Find the following series and their radius and interval of convergence: a) MacLaurin f (x) = 2" b) Taylor at a = 2 , f(x)=x 2x'+x c) MacLaurin f(x) = d) MacLaurin f (x) = In(



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In each part, obtain the Maclaurin series for the function by making an appropriate substitution in the Maclaurin series for ln(1 + x). Include the general term in your answer, and state the radius of convergence of the series. $\begin{array}{ll}{\text { (a) } \ln (1-x)} & {\text { (b) } \ln \left(1+x^{2}\right)} \\ {\text { (c) } \ln (1+2 x)} & {\text { (d) } \ln (2+x)}\end{array}$

Dysfunction in three Our affects to the fourth. You want to find it? Mark Lauren? Serious. You know, you see this? So what we could do is that we could use this function into the X find a much Clarence years for three X on the in there. Just replace that seriously. Place X by extra fourth. So the this year's 43 x now is gonna be because if Mexico's if the X is gonna be this is gonna be if zero positive with the function of zero. I'm sex. Plus the second directive on the function off sex or zero I'm sexy squared, Divided by two was the 30 relative on the function. But, like three factorial 10 6 cube and there were all these critics are zero. So that whoa more times should have the expansion for E to the X. He's gonna be one. Well, that's one lost six loss X squared divided by two factorial waas exist there by three tutorial. That's the one. So this would be we replace. Expects the fourth. She will be one neatly x. Same sex to the boy reflects the fourth will be bomb blast sex to the port lessons to the fourth square, which is still fourth place. It's the force of its today. It's the fourth time Sex the fourth So x 38 But I would like to factorial plus more extreme Well, this is X to the fourth, the third power, which is equal toe to the fourth damn sets forth. I'm 64. You go back to Soil and extend would be X to the fourth. And that also 16 opens for 16 on goal. It's a one. So long. So that is it Mantra groceries and they roll the radius of convergence. Yeah, so this serious has the form. The form of terms is X to the power. It's good for the power and divided by name sartorial. And so, um, I would like to know wind when that converse is. And so, um so for that we could do our You can see the limit of the, um this is the term eight event. The limit of 10 87 plus one over this event. Uh, well, with this is, um, with the this is, uh, smaller than one than converges. So this term is gonna be well x today one over in plus one tutorial and sex today or mhm plus one on one over a seven would be in Austria divided by extra four. And so this time is gonna be Do you have to compute this limit saying goes to infinity? Mhm. If this is satisfied, they're in complete convergence. So, um so So that canceling these four in constant with that so that you could help Excellent Fourth and then in plus one councils in plasma factorial We then So we'll have in plus one. What is the limit off these Or I'm going to pretty well. So four x fixed. This limit goes through zero harada you no matter what is the value of X converges. So where use of convergence? He's, uh so you can't for any X converses. And so well, now we're going to use, uh, Is that information the pain the MacLaurin series for? It's the third and see again, thanks to the fourth. So one thing that you could do is that you could do you could use the far, but we know that what iss the serious for that which is which is this? And then we just multiply the polynomial she would be one last six to the fourth. A sex that it mhm invited by. Yeah, Factorial plus six today. Yeah, well, while we're bye bye. Three factorial my 16. 16. What about four pictorials and multiply these two we'll get, um It's the third 67 for the street at seven. But for 60 day 11, United States from Colonel Plus, um, looks to me three plus 12 through 15. Right, three factorial flash extreme 19. Mhm. Very, very poor factorial on so on. But another method that we have to use is that well, notice that these is the is the internal off if you have, ah, differentiate with depression. Uh huh. Or off extra for Then we're gonna get that get so that Oh, you get that chance. So peaceable toe four four x the third materials before. So what? What you can do, alternatively, is that we could do You could differentiate term by term, Right? Because the ladies of Converse's is infinity. 1/4. Um, so, uh, this serious that we already have this serious effects to the fore. So serious. Is these just just this part? So, uh, that one plus extra four plus, um x 38 divided by two factorial plus six, three, 12, 3 material plus x the 16th by four factorial and so on. Eso if we might be differentiate, that would be the same. That fourth political 10 The tip of this will be or looks the third power. There will be eight. I'm six today. Seven forward. Bye bye. Two factorial. Uh, but off this time would be, uh, 12. Thanks. Today, um, to the 11. Divided by three fucked oil plus 16. I'm 63 Uh, yeah, divided by four factorial. And so one so well, if we cancel those terms, we'll have That's an extra third and then eight bye bye for the issues. Probably about two factorial, that is, that would be one reserve so or so 12 divided by two is three on three divided by three factorial. This would be just two factorial. So yeah, it's a 11 plus 16 valuable voice for on Dana for, by the way, for factorial that is three factorial to be like this and then more terms and so well, this these serious is gonna be extra third plus 67th hour plus 11 next to the 11th, Powered by two factorial was 60 day 15 power divided by 3 ft cereal. Eso won, uh, so if we compare, we see that Have those people hunt those tools? It's much so these ones, much as we'd all with this one because, yeah, both we can do both methods. We can either differentiate this one term by term or we could alternatively do the serious expansion for this one on multiplied by extra third. That gives us the same result.

In this question, we have port a where X were X minus One is equal to negative X over one minor sex, one negative x not the play 1/1, minus sex we choose Equals negative X Milton Fine. Last eggs, plus exports to and so on. So are is equal to one or part B. We would replace X rays export toe. So we will have three plus three over bacterial toe export for los three were bacterial for export beads on so on. So far in this space equal Boston eight for Mr Infinity. Uh, see, in this import see? So we've m equalled. Negative story. So one glass, thanks. On board Negative city is equal one minus three X loves six export toe minus then export city. And so so with replacing X trees to acts, we can you get that x by one last well X or negative city equals X minus six Exports to plus 24 exports three and so on. So in this case are is equally off

And find the 1st 490 terms with this McLaren concurrent series for this function followed by the power series representation in the interval of conversions. So let's get started by comparing this two Kasich of X. That's equal to since one plus X squared over two factorial. That's next to the 4th of our four factorial. Next to the 6 36 Victoria. So when we substitute in three x. and cracks You get one plus three x squared over two factorial Plus three X to the 4th over four pictorial Plus three X to the 6th Over six Victoria. So that submission would be from okay called Syria to divinity three x to the two K two K factorial. So that's why the internal convergence. Next take the limit has and approaches infinity. Uh huh reacts two and plus two 200 plus two factorial It's going to be Times two and Victoria over three X to the two end. So that simplifies to three X squared over two and plus two there's two and this one ticket demand. It has been a purchase infinity that equals zero. It's always less than 12. Therefore interval of convergence From eight infinity to infinity.

Want to find the first four terms of first for nancy returns MacLaurin series followed by the power surges representation and its interval conversions so Can't derivatives so that's going to be three X and Eleanor three 2nd relative would be three X to the island of three squared chain role and we take you on that one more time we have three X. Ln of three to the third power Plugging a zero We've got one Ellen three three squared Then Helen 3 to the 3rd power. Okay, so that leaves us with the terms of it's going to be one Plus packs a lot of three plus alan of three squared X squared over two Spellman of three cute execute over three factorial. All right. So Those are the 1st 4 terms there. So that next give us in some issue for him. 3.5 from Cool. Zero infinity of X. Okay, Alan of 3 to the K We'll break a factorial so that's how we should there. Yeah. Supply the ratio test to find out the interval conversions So plug in it plus one and then time's up reciprocal what's N factorial over to the end on a 3 to that simplifies to X Time settling of three Over at this one. The limit as an approaches infinity Costa zero which is us to one therefore converges from negative infinity to infinity


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