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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.$f(x) = \cos x$

## $\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}$, $R=\infty$

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okay. So fundamental or a serious forever Backs used death and nation that McLaren Siri's assume that has the power. She respond to notion that not show that are still in the backs Goes to zero. Also found associated readers of convergence. And here we have at Max equals Coulson necks. Okay, Sylvan. The first fine cousin zero, which is one. And the derivative is money. Sanks as your zero, and it's gonna be minus one. So we're gonna fund that. Here's the pattern of the George have at zero so foreign before insecurity. And in this and is a natural number is going to be one and foreign plus one or four plus three. This is zero four plus two is gonna be next one. All right, showing. And I use this to do, like, over MacLaurin series for FX silver by definition, this is thie instituted of F zero over a pictorial Times X to the power in. So this gonna be We're plugging this song so it's gonna be like X minus X square with you, Victoria. Plus, that's for over four Victorian minus extra six over Vittorio class and so on and so on. This is actually say case from zero to infinity that you wanted Power K and X to the poor of two K over to K. Victorio. So one case zero, This is positive. One and no, actually, this is all too k. Plus, I'm sorry. Here is one is not ass, though. Yes, this to Kay and Victorio. Yeah, this is over. Final results for the metal Orin Siri's of FX and the readers. Convergence is just the choral line.

University of Illinois at Urbana-Champaign

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