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Which of the following gives a power series representation of the function flz) cos(22?) centered at 0?E(-1)"2*+1 I4n /2 OA (2n + 1)!IAn | 2 (-1)"2 (2n + ...

Question

Which of the following gives a power series representation of the function flz) cos(22?) centered at 0?E(-1)"2*+1 I4n /2 OA (2n + 1)!IAn | 2 (-1)"2 (2n + 1)!Iin ? '(-1)"24n+2 2n + 1)!-1)"22n (2n)!-1)"24n (2n)!

Which of the following gives a power series representation of the function flz) cos(22?) centered at 0? E(-1)"2*+1 I4n /2 OA (2n + 1)! IAn | 2 (-1)"2 (2n + 1)! Iin ? '(-1)"24n+2 2n + 1)! -1)"22n (2n)! -1)"24n (2n)!



Answers

Find power series representations centered at 0 for the following functions using known power series. $$f(x)=\frac{2 x}{\left(1+x^{2}\right)^{2}}$$

Threes An equal to 1 to infinity of -1 to depart and time to negative one up on end. So here in the execution which of the threes are alternating. So here you can see that let a sub and equal to 3s an equal to 1 to infinity of negative one to the power and time to negative one upon. And so here we have to subdue the value of and so here we put an equal to one. So we get a piece of one equal to 91 to the power one. Time to negative one Equal to 91. And when we put an equal to two, so we get is up to equal to 91 to the power to time to 91 of 1. 2. Time bothered you three upon to When you put an equal to three. So we get a sub three equal to 91 to the power three times to 91 upon three Equal to £95 3 An equal to four. So here we get E. So four equal to 91 to the power for time to negative one upon four. Equal to seven upon four one. About unequal to five. So here we get eight of five equal to 91 to the power five times to 91.5 equal to 99.5. So here you can see that threes Unequalled 1 to Infinity of one. Sorry, 91 to the power and time two naidoo one of on and Equal to 91 Positive three upon to 95.3 positive sound upon full 99 Upon five bothered you and so on. -1 to the power and Time to 91 upon end plus so on. So here you can see that the term of the given trees alternate in their science. So the given trees is and alternate alternating. So I threes so our final answer. It alternating. It is a final answer.

Problem Number 58 The Given Siri's submission Off toe. Okay, Multiply X power to keep plus one from K equals zero till Infinity. The Power Series Representation of one minus x Old part negative one is one plus x plus X power to Los X Power three and so on equals to the submission of X Power from K equals zero till infinity. If we replace X by two x squared above, we get the given Syriza's summation off to key Party Multiply X power to keep plus one from key equals zero till infinity, which is equal to one minus two x squared. Our negative one can then but he right this equation Toby equal toe X over one minus two export to and hence the function represented by the power Siri's is X over one minus two X square

Function f X include you and on the square root of one minus x square. Here, notice that we can really stand you and, uh, half. And then under one minus X squared and notice Done. The emperor him the X and we caught your half times. Ah, again, the one my minus X square. And at the time we have the minus two angst and therefore we get echo to the minus X over one minus x square. So we need to stick to this form now and record up if we have now 1/1 minus X the serious represent for this phone. We were submission on the expo. Okay. From such infinity and it's funded for the absolute of ex mother than one now. Well, first of all in it together one minus x square here. So from here we can conclude that no 1/1 minus X square were recruited. A submission from such infinity x two k and X square smarter than one doesn't will be the same is an absolute of EC smaller than one. Now also, the next time we need to get the money. Thanks hair. Do you get that we will multiply everything by the minus. Thanks, Nephi. Well minus X over one minus x square. You go to the minus submission from such infinity and we have and, uh thanks. I owned it to K plus one. And then we will have the forget for absolute of X Monnet and one. And now the next time we will do the ah, we were integrating beside. So then we should get under Clinton Signed now, because I'm this form here. We should get back to, uh uh, just form now. So we'll be, uh, which we get back to the, uh uh huh. And then off the square root off on, uh, off one minus. X squared isn't exactly a good you know. F X now and now, on the right hand side, we should get echo through the minus a mission. And then until the rift er done, We Could you explain what you gave us to the venue, mate? You gave us to from such infinity blessem constant c now and it's ready for the absolute of ex model on one. And here notice that I would check you find a C. We need to let the X equals zero on, uh, on the left hand side, him got equal to zero. And on the right hand side, we still getting that uses a row as well. Plus c information imply c, which is there are so it listen means that the f X echo to the island off the square root of the one minus X square was a coaching. Remind us so much from such infinity ex about you cables to over two cameras to its full it for the X between the A minus one and the one he noticed them when x it could you both minus 11 We have this one will be the city will be divergent. Therefore, that's what we're trying to answer here.

The function f x encouraging a tension in verse under for X square. You notice that the m prime the X go to 1/1 glass 16 expel for so we need to stick to this form now. Pretty calm down and have the 1/1 minus x. The series for his function will be Submission of the expert came from Serge Infinity is really it for the absolute of X more than one. So now we need to create the denominator. Now, if I wanted to replay this ex hand by the minus 16 expert far so well, there's something from this angle this exodus Well, and it means that one of the one blessed 16 expo far it will ico Judah summation off the minus 16 Expert for our K from search infinity. Therefore, we get any coaching A summation off the menace. One power, okay, Usually can run that. I said to about four. Ok, expect for okay. And then we have here were being ah, uh from Serge Infinity. And it's valid form the minus 16 x Well, for smaller than one. Something for the ah, absolutely. The expert for smaller than ah, no 16. So there's something for the X. Absolutely x moaner Dan no. To So from here we want to integrate on both sides. And then under the second sign him. We had examined the function f X and it could you that tension in verse under four X square. On the right hand side, we have the minus one. Okay, do about four game expel for gave us Kuangdi, running by the fog gave this one person constant. C now is fully for the absolute of tanks. It's mounted into you. Find this value. See here we need to let the X equal to zero and tension in Rosero Echo ju zero. So therefore we get a little psycho. 20 on the right hand side began Echo. Jealousy only therefore doesn't implies that we have Seiko Jizo. And a final answer for this one would be Don't in inverse under Far Square, was it? Coggiola summation from Serge Infinity minus one power. Okay, Jimbo. Far. Okay, expel for guy. The violin, my love for guy. Uh, that's one. And then it's valid phone. Uh uh. Ex between, uh, formalist studio here


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