Sum of power series $\sum_{k=1}^{\infty}(-1)^{k-1}\frac{x^{k+1}}{k(k+1)}$

Question

Calculate the sum of power series: $$\sum_{k=1}^{\infty}(-1)^{k-1}\frac{x^{k+1}}{k(k+1)}$$ I know that $$\ln x=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}(x-1)^{k}}{k}$$ but I don't know if and how can I use it to calculate the given sum.

Answer 1

Hint:

Consider the sum you're given. Take the derivative of it with respect to $x$, and note that we also have this series:

$$\ln(1+x) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}x^k$$

Answer 2

An alternative to differentiation is simply to use $\frac{1}{k(k+1)}=\frac1k-\frac{1}{k+1}$, with $$\sum_{k=1}^{\infty}(-1)^{k-1}\frac{x^{k+1}}{k}=x\ln(1+x),\quad\sum_{k=1}^{\infty}(-1)^{k-1}\frac{x^{k+1}}{k+1}=x-\ln(1+x).$$