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Math Help - Log of series

  1. #1
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    Log of series

    Hi everyone...

    Recently I resolved a problem using this equation (stopping to the 2nd term):

    <br />
\ln(1+\alpha_1 x + \alpha_2 x^2 + ...) \approx \alpha_1 x + (\alpha_2 -\frac{1}{2}\alpha_1^2) x^2 +...<br />

    Problem is that it's new to me and I cannot prove it!!

    Would someone, please, give me direction to a web site where I can find a proof?
    Or, eventually, prove it for me here?

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by paolopiace View Post
    Hi everyone...

    Recently I resolved a problem using this equation (stopping to the 2nd term):

    <br />
\ln(1+\alpha_1 x + \alpha_2 x^2 + ...) \approx \alpha_1 x + (\alpha_2 -\frac{1}{2}\alpha_1^2) x^2 +...<br />

    Problem is that it's new to me and I cannot prove it!!

    Would someone, please, give me direction to a web site where I can find a proof?
    Or, eventually, prove it for me here?

    Thanks in advance!
    The following is called a Maclaurin series:
    f(x) = f(0) + xf'(0) +x^2 f''(0)+...........

    f(0) =  \ln(1) = 0
    f'(0) =  \frac{\alpha_1+2\alpha_2 x}{(1+\alpha_1 x + \alpha_2 x^2 + ...)}\bigg{|}_{x=0} = \alpha_1

    f''(0) =  \left(\frac{\alpha_1+2\alpha_2 x}{(1+\alpha_1 x + \alpha_2 x^2 + ...)}\right)'\bigg{|}_{x=0}

    I dint compute the last co-efficient... But it should turn out as you want...
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