$\displaystyle \log(1-x) = -x-\frac{x^2}{2}-\frac{x^2}{3}-\frac{x^4}{4}- \ldots -\frac{x^n}{n} - R_n $ for $\displaystyle -1 \leq x < 1 $.

Why isn't this the case for $\displaystyle -1 < x < 1 $? Instead, why do we replace $\displaystyle x $ with $\displaystyle -x $ to get $\displaystyle \log(1+x) = x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4} +\ldots + (-1)^{n-1} \frac{x^n}{n} - R_{n}' $?