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Find & powver series representation for the {unction_ (Give vour power series representation centered BI * = 0.)Deremine tne Intenval 0' convergence (Enter...

Question

Find & powver series representation for the {unction_ (Give vour power series representation centered BI * = 0.)Deremine tne Intenval 0' convergence (Enter Your anster Using interva nolalion:

Find & powver series representation for the {unction_ (Give vour power series representation centered BI * = 0.) Deremine tne Intenval 0' convergence (Enter Your anster Using interva nolalion:



Answers

Find a power series representation for the function and detemine the interval of convergence. $$f(x)=\frac{1}{1+x}$$

Yeah. What do you call that? You were the 1/1 minus thanks. We could get a summation off the expo. Okay. From search Infinity. And it's valid for the absolute of exponents on one. And then yeah, we will have. No, they will be Bayliss X here by the minus x squared. And then we should get the new function. Now under F X, you go to 1/1 plus X Square and were coaches submission from Sergio Infinity minus X squared. OK. And it was simply together in a submission from zero to infinity minus £1 k expert to gay. And it's valid fonder minus X squared, absolutely smaller than one usually given to the absolute of X moment and one and therefore we found in the series represent for the function F X and the interview under convergence here

Given the agent x ico Judas. Expel que algo que for tire from the infinity and a dozen will be valid for the expert U minus. Infinity! Infinity! Yeah, We want to find a function f x Go to the about your ex. So we need to do is go dependence expert that you anc's something here square And then we should get in close to the submission And then we have that you x okay. Defending because attire. Then we get a coaching. That too. Okay. Expel came over came from tire as they voted for the do X between the months Infinity to infinity And it means that the X must baby two months Infinity, infinity

These represent this as a power series and then find it Bridget It's internal. And so First we need to get this into the form of 1- are here. And so we're going to manipulate this little bit by doing X over one minus negative two X squared. We'll have like this that means our is negative. So this is our our here and then that's for a laptop here. Okay. So we have the summation from an equal zero to infinity of X times negative two X squared to the end of power. So that would translate into The submission from n equals 0 to infinity of -2 to the end times X two to it plus one. Yeah. So let's use the ratio test to be about the interval of convergence. We'll have negative two to the end plus one next to the times X to the two N plus three Over a -2 to the end Times X to the two n Plus one. Okay, so take the absolute value their out of here. left with two x squared. You must end one here. Okay. Means that X squared is less than one half studying taxes between One over radical two. Single too radical too of her too. So that means our interval of convergence is going to be a firm radical two or 2 Negative radical to over to medical to over two and the police won't work for that. So because some corporations test. So this is our interval of convergence

And the internal convergence. Do that. 1st. Take the limit as approaches infinity. We're doing the ratio. Test this. So we have data want and plus two plus two times extend that plus one Divided by a -1 to the plus one Krampus one. To the extent that here and so supplying this here, we can ignore the negative ones because that's absolute value. The endless to an endless one. Go to one when we take into infinity. And the extra bands here pieces with just X. So our potentials from negative one 21 But we need to test the endpoints here. So let's take a look at that. You look at x equals negative one. We get -1 to the end plus one Times and Plus one times negative 1 to the head. So this will diverge. Sure, excellent. To that just goes to infinity. Just to keep going and going and going. If we take a look at X equals one -1 to the end plus one Some Plus one times 1. It also diverges because that's also getting bigger and bigger, Right? So that means our interval of convergence. That's your negative 1- one.


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