Q. $\frac{1}{2} - \frac{1}{{2.2}^{2}} + \frac{1}{{3.2}^{3}} - \frac{1}{{4.2}^{4}} + \ldots$ is equal to
A) $\frac{1}{4}$
B) $\log_{3}\left( \frac{3}{4} \right)$
C) $\log_{e}\left( \frac{3}{2} \right)$
D) $\log_{e}\left( \frac{2}{3} \right)$
Explanation
(c)
On putting $x = \frac{1}{2}$ in $\log_{e}{(1 + x)} = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + ...\infty$
We get, $\frac{1}{2} - \frac{1}{{2.2}^{2}} + \frac{1}{{3.2}^{3}} - \frac{1}{{4.2}^{4}} + ... = \log_{e}\left( 1 + \frac{1}{2} \right) = \log_{e}\left( \frac{3}{2} \right)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.