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Math Help - Taylor series of erf

  1. #1
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    Taylor series of erf

    Hi,

    When I use substitution to find the power series of e^(-x^2) at x=0, I get 1-x^2+x^4/2!-x^6/3!+...

    However, when I take the derivative of e^(-x^2) directly, I get: 1-2x^2+12x^4/2!-...

    Can anyone tell me why this is the case? Did I do the derivative wrong?

    Thanks in advance
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  2. #2
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    Re: Taylor series of erf

    y(x) = e^(-x^2) = 1-x^2+x^4/2!-x^6/3!+...
    y' = ?
    First method :
    y' = (1-x^2+x^4/2!-x^6/3!+...)' = -2x+4x^3/2!-6x^5/3!+...
    y' = -2x+2x^3-x^5+...
    Second method :
    y' = -2x e^(-x^2)
    y' = -2x (1-x^2+x^4/2!-x^6/3!+...)
    y' = -2x+2x^3-2x^5/2!+2x^7/3!+...
    y' = -2x+2x^3-x^5+...
    Both methods lead to the same result.
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