This problem can be solved using the integral method. $$\begin{align*} G(x)=\ &x-\frac{1}{2}x^2 +\frac{1}{3}x^3 - \frac{1}{4}x^4 +\cdots \\ =\ &\int (1-x+x^2 - x^3 +\cdots)dx\\=\ &\int \frac{1}{1+x}dx \\ =\ &\ln(1+x)+C \end{align*}$$

where $C$ is any constant which can be determined by setting $x=0$: $$G(0)=0=\ln(1+0) +C\implies C=0$$

It follows that the answer is $\boxed{\ln(1+x)}$.