Adding and Subtracting 1 on each term
$S= \frac{1+x -1}{1-x^{2}}+\frac{1+x^{2} -1}{1-x^{4}}+\frac{1+x^{4} -1}{1-x^{8}}+\frac{1+x^{8} -1}{1-x^{16}}$
Using $(a^{2}-b^{2})=(a+b)(a-b)$
$S=\frac{1}{1-x}-\frac{1}{1-x^{2}}+\frac{1}{1-x^{2}}-\frac{1}{1-x^{4}}+\frac{1}{1-x^{4}}-\frac{1}{1-x^{8}}+\frac{1}{1-x^{8}}-\frac{1}{1-x^{16}}$
Simplifies to $S = \frac{1}{1-x} - \frac{1}{1-x^{16}}$
Answer: C