1. ## Log of series

Hi everyone...

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

$\displaystyle \ln(1+\alpha_1 x + \alpha_2 x^2 + ...) \approx \alpha_1 x + (\alpha_2 -\frac{1}{2}\alpha_1^2) x^2 +...$

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?

2. Originally Posted by paolopiace
Hi everyone...

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

$\displaystyle \ln(1+\alpha_1 x + \alpha_2 x^2 + ...) \approx \alpha_1 x + (\alpha_2 -\frac{1}{2}\alpha_1^2) x^2 +...$

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?

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