Q. If $|x| < 1$ and $y = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + ...$, then $x$ is equal to
A) $y + \frac{y^{2}}{2} + \frac{y^{3}}{3} + ...$
B) $y - \frac{y^{2}}{2} + \frac{y^{3}}{3} - \frac{y^{4}}{4} + ...$
C) $y + \frac{y^{2}}{2!} + \frac{y^{3}}{3!} + ...$
D) $y - \frac{y^{2}}{2!} + \frac{y^{3}}{3!} - \frac{y^{4}}{4!} + ...$
Explanation
(c)
$y = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + ... = \log(1 + x)$
$\Rightarrow 1 + x = e^{y} = 1 + y + \frac{y^{2}}{2!} + ...$
$\Rightarrow x = y + \frac{y^{2}}{2!} + \frac{y^{3}}{3!} + ...$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.