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Find the first three terms of the Maclaurin series for `xe^(-x)` Consider that it is...

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juniorsilvamath | Student, Undergraduate | (Level 1) Honors

Posted March 10, 2013 at 7:17 PM via web

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Find the first three terms of the Maclaurin series for `xe^(-x)`

Consider that it is centered at x = 0. 

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted March 10, 2013 at 7:55 PM (Answer #1)

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Maclaurin series of infinitely differentiable function `f` is defined as

`f(x)=sum_(n=0)^infty(f^((n))(0))/(n!)x^n=f(0)+(f'(0))/(1!)x+(f''(0))/(2!)x^2+cdots`` `

1st term

`f(0)=0e^0=0` <-- First term

2nd term

`f'(x)=e^(-x)-xe^(-x)`

`f'(0)=e^0-0e^0=1-0=1`

`1/(1!)x=x`  <-- Second term

3rd term

`f''(x)=-e^(-x)-(e^(-x)-xe^(-x))=-2e^(-x)+xe^(-x)`

`f''(x)=-2e^0+0e^0=-2`

`-2/(2!)x^2=-x^2`  <-- Third term

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