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Find the first three nonzero terms of the Maclaurin expansion of the given functions.$$f(x)=e^{-2 x}$$

$1+\frac{-2}{1 !} x+\frac{4}{2 !} x^{2}+\frac{-8}{3 !} x^{3}$

Calculus 2 / BC

Chapter 30

Expansion of Functions in Series

Section 2

Mclaurin Series

Series

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we want to find the first three non zero terms of the MacLaurin series. For the function F of X is equal to eat the negative two X. This question is challenging understanding to expand a function using a series in particular is testing our knowledge of MacLaurin series, which states that the function F of X is equal to f zero plus F prime zero X plus F double prime 0/2 factorial X squared plus F three of 0/3 factorial X cube and so on. Where every term of the MacLaurin series is the derivative F zero divided by n factorial times X. To the M. So even the negative two. X zero is one. We have to find at least the first two hundreds of fx in order to find the 1st +390 terms. Thus, we have F one equals negative to either negative two, X zero is negative two. F two is four, even negative two X at zero. This is for either zero or four. Thus, from F F one and F two, we found our 1st 390 terms and we can plug into the series. That's our series is one minus two X plus two X squared, and so on.

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