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Math Help - taylor expansion of erf(x)

  1. #1
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    taylor expansion of erf(x)

    how do you solve:

    consider the so called error integral erf(x) for x ∈R:
    erf(x)= 2/ (pi)^(1/2) integrate exp (-y) from x-x to x=0.
    derive the first few terms in taylor expansion of erf(x) up to x^3. hint: use second fundamental theorem of calculus

    can someone help me solve this? im completely lost..
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    how do you solve:

    consider the so called error integral erf(x) for x ∈R:
    erf(x)= 2/ (pi)^(1/2) integrate exp (-y) from x-x to x=0. ???
    derive the first few terms in taylor expansion of erf(x) up to x^3. hint: use second fundamental theorem of calculus

    can someone help me solve this? im completely lost..
    erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt

    using the 2nd FTC ...

    erf'(x) = \frac{2}{\sqrt{\pi}} e^{-x^2}

    note that e^u = 1 + u + \frac{u^2}{2!} + \frac{u^3}{3!} + ...

    so ...

    e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \frac{x^6}{3!} + ...

    integrate term for term ...

    \frac{2}{\sqrt{\pi}} \int e^{-x^2} \, dx = \frac{2}{\sqrt{\pi}} \left(C + x - \frac{x^3}{3} + \frac{x^5}{5 \cdot 2!} - \frac{x^7}{7 \cdot 3!} + ...\right)

    finish up ...
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  3. #3
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    do i need to find the remainder since this is a taylor series question? or what else do i need to do to finish up?
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  4. #4
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    Quote Originally Posted by alexandrabel90 View Post
    do i need to find the remainder since this is a taylor series question? or what else do i need to do to finish up?
    the directions state ...

    derive the first few terms in taylor expansion of erf(x) up to x^3
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