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Find the Taylor series for:A. f(z) = x centered at x =-1B. f(z) = sin(x) centered at € = T C. f(s) = e centered at % =1 D. fl(z) = ze2c centered at z = 0 E. f...

Question

Find the Taylor series for:A. f(z) = x centered at x =-1B. f(z) = sin(x) centered at € = T C. f(s) = e centered at % =1 D. fl(z) = ze2c centered at z = 0 E. f(z) = zcos(z?) centered at z = 0 sin(z) F. f(z) centered at % = 0G. f(c)centered at x = 0 1)2Hint: Start with theMaclaurin series ofH: f() centered at x =1 Hint: Rewrite the 2x 12 denominator you Cal uSC the Maclaurin series of

Find the Taylor series for: A. f(z) = x centered at x =-1 B. f(z) = sin(x) centered at € = T C. f(s) = e centered at % =1 D. fl(z) = ze2c centered at z = 0 E. f(z) = zcos(z?) centered at z = 0 sin(z) F. f(z) centered at % = 0 G. f(c) centered at x = 0 1)2 Hint: Start with the Maclaurin series of H: f() centered at x =1 Hint: Rewrite the 2x 12 denominator you Cal uSC the Maclaurin series of



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 $R_{n}(x) \rightarrow 0.1]$

$$f(x)=\cos x, \quad a=\pi$$

So this seems seem, uh, first derivative. KUSA next second derivative negative sine x no third derivative. Negative. Go sign X fourth derivative. Get back to sign X. Right. So therefore, derivative, get back to the sine X and then it continues the pattern. Still going to go through this one again and continues as a pattern right now you have pi over two. This is zero. This is one zero, you know? So that is the same thing. Negative one zero. Right, So? So that is that is Ah, the pattern that you can see here. So what is happening? If you look at this one, you can see a pattern. You see? It zeroes once and negative ones. Right? So is Tomblin struggling? So is tabling Oh, so this is first derivative. Second derivative third derivatives, fourth derivative. Right? Uh, so this is the anti derivative, right? The the root of violated at that. This negative won t in, right? It is negative One city. Ah, Okay, so we have to split it because someone some some are zero. So that split it some other 07 Someone Not exactly. So okay for ah, for even powers for even powers off derivatives. Right? Uh, even powers. I think I made a little mistake. What is? Sign off pi over two. That is actually one. It is rather cosom power to that zero by the opposite. So if I put pirate to here, coz and power to is zero right, So here is supposed to be zero and ones and zeros and ones. So see you see here. So this is zero now what is negative? Sign Pirate tatoos. Negative one Negative co sign zero positive sign piled with you one. Okay? Yeah, that is actually what it is. It is Costa Empire, that is zero Khost Empire reduce zero and signed. Probably two is one. Right? So that is what you have. Okay, so for even powers off derivatives. Easy. This is at the zeroth derivative. This is it. A second derivative inside of fourth derivative. So for even powers off derivatives, you can see that the function is so for even powers. Does Iran even powers free? Even Paris? That function is negative one. It is troubling between wanted negative one. Right? Because here's one here is negative one here is one. So for even powers, it toggles between negative one in positive. And that is why you have a negative one to the power in right todo to depict that one and then for art powers. So there's a right odd for odd powers. It is zero. Right. Our powers. This one here is odd. Third derivative is odd, you know? So zeros for our powers. Good. So what is a Taylor series expansion then? Ah, so you know, we already have the, uh We've written the formula from the territories a lot of times, So if you follow the tutorials diligently, uh, you see, So when you put everything in there, you're gonna get a terrorism use off one minus. Uh, what have ah, 1/2 factorial, right? Uh, X men despise X minus pi over two squared I That is from the first derivative. Ah, So this is, uh, the zeroth derivative. Second derivative? Because for even powers, you have something for our powers. You have nothing because the our powers is zero. Right? And then you're gonna get, uh, because final four factorial X minus pi. Would you to the pound for right? You get any parent? You know, it is just this ones are regions for even powers of the derivative. Right? For our powers were getting zero for even powers were getting negative one and one. Right, So that's what is happening. That is why I just free discarded the own powers. And I'm just writing the even power because the out of powers or zero so is gonna continue front forever, right? For different, presentable future. And I mean, for the ants derivative for the 10th derivative, right, this is the even power and derivative, right? So for the anti derivative, we're gonna get, uh, negative one today in Is this one that I've written? And then if you look here, you see that this one is to you. This one is for the other one is gonna be six factorial. So it is increasing in the powers it is increasing in multiples of two. So this is going to in factorial and we just reading a, uh a, uh a pattern free. And this one is gonna be expires. Pi over two and the powers of increasing multiples of two. This is two. This is for the other one is going to six, right? So you can just write to in here. So there is There is a parent, and then is that it can continue for the foreseeable future. So or is gonna continue into infinity? That's what I'm trying to say. Eso Finally. Then it's gonna be summation and from zero toe an right and this is gonna be the parents. So you just gonna put that one there, right? For all in, Right? So that is gonna be over two and factorial X minus a pi over 2 to 2 in So once you have that. Ah, yes. So, ah, you can go on to find the radius of corporations or whatever means that a problem. So that is the answer. That is the Taylor series expansion.

To start this problem, we're going to calculate as many derivatives of the coastline of two X that is needed such that when we plug zero into them their non zero, we have four of them that are non zero. So we start with our function, just people to you. The co sign of two X. We take the first derivative. This will be able to using the chain role. Take the derivative of the co signer two x, which would be a negative sign of two x kinds of derivative that the inside which will just be too to the overall derivative of the negative to sign or two X The second derivative will follow the same format, which would be negative multiplied by 24 co sign of two x The third derivative of X will be the derivative of coastline is negative sign. So this one this one was positive. Eight. Sign of two X. Now we can plug in or a zero. Do you have the co sign of two times zero which is zero the coast and zero is one first derivative zero negative to sign of zero sign of zero is zero. So this is a zero term, so we're gonna need to take extra derivatives here. Second derivative at zero would be negative for terms. The coastline of zero, which would be negative four times one and the third derivative at zero will be eight times the sign of zero, which is zero. So we're gonna need to take a few more derivatives. The fourth derivative of X, we'll be 16 co sign of two X using the chain rule and the fourth derivative at zero will be 16 co sign zero Just be 16. The fifth derivative of X will be negative 32. Sign of two X plucking zero into that will give negative 32 times the sign zero, which is just zero. You take one more derivative, which is the sixth derivative, which is negative. 64. Co sign two x bring in zero. Into that, we'll get negative 64 from signing zero, which is negative. 64 Now we can use all of these to write out the first terms in the Taylor Siri's B F A, which was just one plus f prime of a Times X minus a, which is zero plus half double primavera which was negative for over two factorial X minus A. But a zero sum should just be X Square plus F triple prime of zero is just zero plus the fourth derivative zero, which is 16 over, and she'll be four factorial times x to the fourth power. Plus, the fifth derivative of zero is zero plus the six derivative zero, which negative 64 over. In this seventh terms will be six factorial times next to the six Buster. These were the first for non zero terms and we can write this out in a way such that we try to see a pattern and I'm gonna use the fact that one is 2 to 0 of power for these two to the second Power 16 is to go to the fourth power and 64 2 to the six power and also used the fact that one is also zero factorial, so we can rewrite this as 2 to 0/0 factorial minus two squared over two factorial x squared plus to the fourth over. Four factorial extra the fourth minus to to the 6/6 fact Orio Extra the sixth And now we can write this in its general form as a Siri's from K equals zero to infinity. But first, I'm gonna take care of the alternating plus and minus signs by putting in a negative one to the K of power. Since we start positive and in the first, the second term is negative. Negative onto the K takes care of that times two que zero we get to zero. If K is one, we have to get a two in the exponents here just will have to be a two times K in the exponents. In the denominator of a coefficients, we have zero factorial and then two factorial and then four factorial. So we'll also need to k factorial times x first term, the extra zero an X squared exports. This will be an extra two k. Another way you can rewrite this is by using the fact that to to the two k is two squared to the chaos power and therefore you'll have negative one to the cave power and then a Ford's in the cave power. It's just just together negative for to the caves power over two k factorial times X to the two k and this is the Taylor Siris for the coast. Time to x at equals zero. Now we can look into the interval of convergence by taking the limit Has que person affinity of the absolute value of a K plus weren over a sub k where this is our ace, okay. And this will be the limit as que person infinity of the absolute value of negative four to the K plus one negative force to the Cape Osborne is negative for to the k times negative four times x to the two k two times k plus one over to terms K plus one factorial times the reciprocal of a sub K which is two k factorial divided by negative 42 k s Power Times X to the UK power. And now we can rewrite some terms by expanding out X to the two times K plus one being ableto X to the two K plus two which is equal to X to the two k times X squared. We can also look at two times K plus one. Factorial is equal to two k plus two factorial that you came close to factorial is just two k plus two times two k plus one times two k times to K minus one and so on. In these terms, beyond here are all just cape tu que factorial So therefore, two K plus one factorial is two k plus two times two k plus one times two k factorial. And now we can cancel house in terms. Since we have ah, negative four to the K factor in the top and bottom, we have an extra the two K factor and top and bottom. We have a two K factorial factor, a top and bottom, so we will be left with the absolute value of negative for terms X squared, divided by two K plus two times two K plus one where we need the limit OutFront limit as que approaches infinity and negative four to the X Negative four x squared does not rely on K so we can bring it out in front of the limit to get it. Is this times that limit as K brushes infinity of the absolute value of 1/2 K plus two turns to K plus one and this limit has take goes to infinity is zero. So in the end this limit is always zero. So that means for any choice of X limit will go to zero, which is less than one. So that means X can be any value for the tailor Siri's to converge.

In discussion we? Re calling about the Tyler series. Okay. And it's a stand any function F. X. Can be written as the series on the form and the review of the function evaluated the C. Times X minus C. Bell. And even in the N. Factorial from zero to infinity. In this question were given the function F X echo two the if our X by the sea and we go to the one. Now let's try to find the first few terms on the series. So the first time we were the F. Of the one. So you got even once we got you. E. So the next term it will be the F. Prem. I'm the one. So exactly what you see as well. And every single term here we can expect the F. N. I'm the one. It will go to be as well. So therefore we can get the F. X. Here can be written as the submission of the E times X minus one power. And you're running by and pictorial And start from 0 to Infinity. So we can bring the outside Uganda a term submission on the x minus one hour. And even in the N factorial from the infinity. And this will be the Treasury's on the function

Yeah. In this question will record about the tyrant series. So we have a ton of serious here. Okay. Initially fight as the threes on the farm. F. X. Can be written as a series on the form F. And the crew. The reviews in the in the sea an X. Man see well and inviting my inventory real and start from zero to infinity. And in this question were given the function F X. Is defined as the E Bella two X. And with the C equal to zero. Now look at the formula, we need to fight the end derivative and evaluate and see which is the role. So let's try to go ahead to find the first derivative. So the first one will be the zero here. So which is the function itself. So equal to evil two X. And therefore the after undersea which is zero here with equal to a power zero equal to one. Similarly we can fight the first the river till on the X. Which is equal to the to be about two X. And we're blocking the value of the sea. Which is the wrong. So when you go to the to Eba Israel equal to two and the next one we should have the secondary with you which can echo to the far about two X. So we have the second, he referred to a family under zero. If we go too far even zero equal to far and maybe one more. And the third derivative we go to the uh this one will be the eight E about two X. So we value A And zero. We get equal to the eight times uh zero which is equal to eight. And we noticed that we have some button here So we should expect to get the f. And the derivative very under zero. It must Echo two. Here. We can notice that this one will be in the two hours zero. It is going to be to about one. This will be too bad to be to about three. So here we should get to about. And as a result we can write the F. X. Equal to the evo two X. It will be equal to the submission of the tube end. Ex miners zero well and inviting behind and factorial goes from zero to infinity. Or we can register and to the submission of the two X. Well and inviting by in victoria goes from zero to infinity. And this will be the answer.


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