1. ## Power Series

From the power series,

$x - \frac{x^2}{2} + \frac{x^3}{3} - ... + (-1)^{n-1} \frac{x^n}{n} + ... = \ln (1 + x),$

obtain the power series whose sum is

$\ln (\frac{1+x}{1-x}).$

2. Originally Posted by cgiulz
From the power series,

$x - \frac{x^2}{2} + \frac{x^3}{3} - ... + (-1)^{n-1} \frac{x^n}{n} + ... = \ln (1 + x),$

obtain the power series whose sum is

$\ln (\frac{1+x}{1-x}).$

(1+x)/(1-x) = 1 + 2x/(1-x) ==> simmilarly as the above one we get:

ln(1 - 2x/(1-x)) = 2x/(1-x) - [2x/(1-x)]^2/2 + [2x/(1-x)]^3/3 -....

Tonio

3. $\ln (1+x)-\ln (1-x)=\ln (1+x)-\ln \big(1+(-x)\big).$