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Find the Maclaurin series for $ f(x) $ using the definition of a Maclaurin series. [ Assume that $ f $ has a power series expansion. Do not show that $ R_n (x) \to 0. $] Also find the associated radius of convergence.

$ f(x) = x \cos x $

$f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n) !} \quad R=\infty$

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Okay, so use definition of Mickler serious to find the metal answers for half of eggs. All right, so ever actually goes to extremes, cousin effects both. And the definition for meth or serious is if we expand it at at zero is gonna be It's absurd to Millie after instructive of zero times X to the in order and Territorial. And we know that the readers committees for effort six and the articles to Billy No. And yet the first purity forever. This is going to be a co site x minus like Site X And second, the rhythm is going to be minus sine X minus esko Xanax and third after the after the vaccine is going be minus coz i x plus x science. So let's can you do it? It's going to do it and find some parents off the driver there. So the affects to the four thirty, but it's going to be minus Plus, Sykes purse looks closer. Yes, All right, so here a phone, there's a pattern. And if you even do the after five, there's going to be co signed. X One is a science which is just equals two with one prime of X. So the pattern ISS, they're right on here. So we fund that so half in FF affects to the river, then it's equals two. The first fourth of max. Yeah, and in this greater or equal to one. So we put in zero. Let's find out what this crunch you're There's gonna be one one zero. It's one and double crew empty is going to be zero minus zero minus ministerial student zero and the third through him. It's going to be minus one. And the fourth there there is going to be zero Okay with one net. So, uh, alright, let's read this. What? We first right on the pants for it with acts zero. So life, the foreign press one for us with three in the back. Uh, affect backs at zero is gonna be one and foreign plus two zero foreign car three zero. It's minus one and four plus four, which is just foreign zero is still zero. All right, so we first raised this out and where you see Do you know for mother for vets, the MacLaurin experience effects to find out what is thie middle answers for X Times Co's on Nets. All right, so here, that's going to be so in. From zero to infinity. Uh, that's the power of foreign cuss one over phone, Cass. When pictorial minus extra poor foreign for three for in that story. Victoria. So that should be the Met Lauren experience of attacks at the point X equals to zero. So let's check this were put in the several terms into it. So when any zeros this isthe one and this is this is three minutes. Zero. Because these two terms is zero and which have the denominator, It's just one and three and one one. Any one is will be five and eight and seven, yet which is which is consistent with here. Okay, so that's the final answer for this question. No.

University of Illinois at Urbana-Champaign