$\log(1+x)=x-\frac{x^{2}}{2} +\frac{x^{3}}{3} - \frac{x^{4}}{4}+...$
$\log(1-x)= -x-\frac{x^{2}}{2} - \frac{x^{3}}{3} - \frac{x^{4}}{4}-...$
$\frac{1}{(1-x)}=1+x+x^{2}+x^{3}+x^{4}+...$
$ \frac{x}{(1-x)}=x+x^{2}+x^{3}+x^{4}+x^{5}+...$
Let us check option D:
$\log(1-x) + \frac{x}{(1-x)} = -x-\frac{x^{2}}{2} -\frac{x^{3}}{3} - \frac{x^{4}}{4}-...$ $+$ $x+x^{2}+x^{3}+x^{4}+x^{5}...$
$=\frac{x^{2}}{2} +\frac{2x^{3}}{3} +\frac{3x^{4}}{4}+...$
which matches with given question..
Hence option d is the answer.