Sum of $\frac{x^2}{2*1} - \frac{x^3}{3*2} + \frac{x^4}{4*3} - ...$ I have to find the sum of : 
$$\frac{x^2}{2*1} - \frac{x^3}{3*2} + \frac{x^4}{4*3} - \frac{x^5}{5*4} +\cdots$$
So far I have :
$$\sum_{n=1}^{\infty}  \frac{(-1)^{n-1} \, x^{n+1}}{(n+1)(n)}$$
which is very close to $\ln(1+x)$... but I just can't figure out what I have to do from there.

Hint:
On its domain of convergence, the derivative of the sum of this power series is
$$x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\dotsm=\ln(1+x).$$
