Can someone help me proof that for any $\displaystyle x \in (-1,1)$ we have $\displaystyle ln(1-x) = -\sum_{n=1}^{\infty}{\frac{x^n}{n}}$
thanks
From power series we know that $\displaystyle \frac{1}{1-x}=\sum\limits_{n=0}^\infty x^n\,,\,\,\forall\,x\in\,(-1,1)\,\Longrightarrow -\ln(1-x)=\int\frac{1}{1-x}\,dx=\sum\limits_{n=0}^\infty\int x^n dx$ $\displaystyle =\sum\limits_{n=0}^\infty\frac{x^{n+1}}{n+1}\,,\,\ ,\forall\,x\in\,(-1,1)$