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Math Help - sigma (c_n * x^n) from 0 to inf

  1. #1
    s3a
    s3a is offline
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    sigma (c_n * x^n) from 0 to inf

    Suppose that
    Find the first few coefficients.
    ?
    ?
    ?
    ?
    ?

    Find the radius of convergence of the power series.
    ?

    I get that R = 6 (which is correct) but I don't know how to find c_0 to c_4. What I think I need to do is isolate x^n and then the rest is the c stuff. So I got sigma from 0 to inf of (-1/6)^n * x^n so (-1/6)^n so, shouldn't that be by C_n values?

    This is what I get (which is apparently wrong according to my school's computer system):

    C_0 = (-1/6)^0 = 1
    C_1 = (-1/6)^1 = -1/6
    C_2 = (-1/6)^2 = 1/36
    C_3 = (-1/6)^3 = -1/216
    C_4 = (-1/6)^4 = 1/1296

    Any help would be GREATLY appreciated!
    Thanks in advance!
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  2. #2
    MHF Contributor
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    Quote Originally Posted by s3a View Post
    Suppose that
    Find the first few coefficients.
    ?
    ?
    ?
    ?
    ?

    Find the radius of convergence of the power series.
    ?

    I get that R = 6 (which is correct) but I don't know how to find c_0 to c_4. What I think I need to do is isolate x^n and then the rest is the c stuff. So I got sigma from 0 to inf of (-1/6)^n * x^n so (-1/6)^n so, shouldn't that be by C_n values?

    This is what I get (which is apparently wrong according to my school's computer system):

    C_0 = (-1/6)^0 = 1
    C_1 = (-1/6)^1 = -1/6
    C_2 = (-1/6)^2 = 1/36
    C_3 = (-1/6)^3 = -1/216
    C_4 = (-1/6)^4 = 1/1296

    Any help would be GREATLY appreciated!
    Thanks in advance!
    Is there a differential equation associated with this question?
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  3. #3
    Junior Member autumn's Avatar
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    unfortunalty, im stuck in hicks state. IN.
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    I would use the geometric series.

    {3x\over 6+x}=3x\left({1/6\over 1-(-x/6)}\right)

    ={x\over 2}\sum_{n=0}^{\infty}\left({-x\over 6}\right)^n

    ={1\over 2}\sum_{n=0}^{\infty}{(-1)^nx^{n+1}\over 6^n}
    Last edited by autumn; May 9th 2010 at 12:07 AM.
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