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Math Help - find the maclaurin series

  1. #1
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    find the maclaurin series

    i've gotten part of the way through into this problem and i'm just a little bit confused..

    find the maclaurin series of f(x)= xe^x

    and i can't solve it with a geometric series, i have to use the definition for a maclaurin series itself

    so, i took the 1st, 2nd, & 3rd derivatives, and determined that the pattern was that the nth derivative will be e^x(n+x)

    then i said that evaluated at zero, the nth derivative is e^0 (n+0) = n

    so i plugged it into the formula of the nth derivative evaluated at zero times x^n all over n!

    i said the first few terms were x + x^2 + 1/2 x^3 + 6x^4+...

    now i'm sort of clueless as to where to go from here
    please help!
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  2. #2
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    Quote Originally Posted by buttonbear View Post
    i've gotten part of the way through into this problem and i'm just a little bit confused..

    find the maclaurin series of f(x)= xe^x

    and i can't solve it with a geometric series, i have to use the definition for a maclaurin series itself

    so, i took the 1st, 2nd, & 3rd derivatives, and determined that the pattern was that the nth derivative will be e^x(n+x)

    then i said that evaluated at zero, the nth derivative is e^0 (n+0) = n

    so i plugged it into the formula of the nth derivative evaluated at zero times x^n all over n!

    i said the first few terms were x + x^2 + 1/2 x^3 + 6x^4+...

    now i'm sort of clueless as to where to go from here
    please help!
    maclaurin series for e^x :

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

    multiply by x

    xe^x = x + x^2 + \frac{x^3}{2!} + \frac{x^4}{3!} + ...
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  3. #3
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    i was told not to use anything that we proved in class (i.e. the maclaurin series for e^x) and to start from scratch, that's why i was doing all that work
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  4. #4
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    f(x) = xe^x ... f(0) = 0

    f'(x) = (x+1)e^x ... f'(0) = 1

    f''(x) = (x+2)e^x ... f''(0) = 2

    f'''(x) = (x+3)e^x ... f'''(0) = 3


    f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + ...

    xe^x = 0 + x + \frac{2x^2}{2!} + \frac{3x^3}{3!} + \frac{4x^4}{4!} + ...

    xe^x = 0 + x + x^2 + \frac{x^3}{2!} + \frac{x^4}{3!} + ...
    Last edited by skeeter; April 15th 2009 at 06:01 PM.
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  5. #5
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    okay, so, this is the taylor polynomial for f(x) = xe^x?

    and so then that means that it's true whenever the remainder is zero.. so i'm kind of confused as to how to show that

    and how should i find the radius of convergence?
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  6. #6
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by buttonbear View Post
    okay, so, this is the taylor polynomial for f(x) = xe^x?

    and so then that means that it's true whenever the remainder is zero.. so i'm kind of confused as to how to show that

    and how should i find the radius of convergence?
    Use the ratio test to get the interval of convergence
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