Logarithmic Series

The series $\displaystyle\sum_{n=1}^\infty(-1)^{n+1}\frac{x^n}n$ is called a logarithmic series. It converges for every $x$ satisfying $|x|<1$, and also converges (though not covered by the general theory here) at $x=1$.

For $|x|<1$, the sum of the series is $\log(1+x)$:

Substituting $-x$ for $x$:

Since $\log\left(\dfrac{1+x}{1-x}\right)=\log(1+x)-\log(1-x)$, subtracting the two series (the even-power terms cancel) gives

Rescaling. To expand $\log(1+2x)$: substitute $y=2x$ into $\log(1+y)=y-\tfrac{y^2}2+\tfrac{y^3}3-\cdots$ (valid $|y|<1$). Since $|y|<1\iff|2x|<1\iff|x|<\tfrac12$, the rescaled series

$\log(1+2x) = 2x-\frac{(2x)^2}2+\frac{(2x)^3}3-\frac{(2x)^4}4+\cdots = 2x-2x^2+\frac{8x^3}3-4x^4+\cdots$ …