5 answers

Find a power series representation for f(x) xe*(-l)"_x2n+1 2 (2n)!X+l 2 (n)l( x2n+2 2 n=0 (2n + 1)!None of these

Question:

Find a power series representation for f(x) xe* (-l)"_x2n+1 2 (2n)! X+l 2 (n)l ( x2n+2 2 n=0 (2n + 1)! None of these is correct:


Answers

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

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Our series representation of this year. So first order to get that uh what first like to focus on is that 1/1 plus X is equal to 1/1 minus negative X. Here. So that's equal to the salvation of from an equal zero to infinity of negative X to the power here. So keeping that in mind, if we take the derivative of both sides for that, that's so directive of 1/1 plus X.

It's going to negative one over plus X squared. And so that um is what we have here. But if we want to positive then you need to indicate that. So that means we have to take the derivative of the submission equals zero negative X to the end. So that's equivalent to so that's negative X.

To the And then this one times and times negative one. But because we have it's observation for medical serial to infinity. But because we have this equal to negative 1/1 plus X squared the multiplying both sides, the negative one to get 1/1 plus X squared as we would like Be equal to n equals 0 to community. So this negative one will go away. So P N times negative X yeah, minus one.

So this would be our power series representation..

Power series representation for 1/1 plus X squared equivalent to if we start out with 1/1 minus have sex, that has a power series representation of negative X to the end here. And so then that's from an equal 02 infinity. Take the dread of both sides respect to X. Get over here. So that's negative.

1/1 plus. This is the negative one plus X squared equals Submission from Ankles. 1 to infinity of negative hex In the end -1 times a hand, times negative one. Okay, But we want to get rid of that negative one there because it's positive one that we have there in front our problem here. So multiply everything by negative one.

We get 1/1 plus X squared just what we want. That's equal to uh So negative and negative cancels or equal to the solution of any was 12 For an times negative X. to the N -1. That is harmful. Are serious.

Restitution..

You get to over three has won my sex squared to a power serious representation. So first I'd like to point out that 1/1 minus X opportunity Submission from an equal zero to infinity of X to the head. So then we derive both sides. We're going to get 1/1 minus X squared equals summation from N equals zero to community. And X.

And I just want since we just have two thirds on the outside here being multiply its we're gonna multiply by two thirds to both sides here again to over three times one minus X squared is now equal to two thirds submission of 02 infinity of and types Next to the advice one and that's it. So what's on the right side here at this passers?.

Want to get to over 38 times one minus X to the second power. First point out that we have one For 1 -1 sequel to the summation from an equal zero infinity of thanks to the end. So we took the dread of respect to X. Both sides here. So then we would get Point of her.

1 -1 squared equals salvation of equals one of an times X two and minus one. Well, the objective is to get two thirds there. So then let's multiply by 2/3 and to both sides. So now we got to over three times 1 -6 squared equals two thirds summation of and equals one to infinity. and in terms next to the N -1, that's our paras series representation..

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