Q. The sum of the series $\log_{4}2 - \log_{8}2 + \log_{16}2 - \log_{32}2 + ...$ is
A) $e^{2}$
B) $\log_{e}{2 + 1}$
C) $\log_{e}{3 - 2}$
D) $1 - \log_{e}2$
Explanation
(d)
$\log_{4}2 - \log_{8}2 + \log_{16}2 - \log_{32}2 + ...$
$= \log_{2^{2}}2 - \log_{2^{3}}2 + \log_{2^{4}}2 - \log_{2^{5}}2 + ...$
$= \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + ... + 1 - 1$
$= 1 - \left( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ... \right) = 1 - \log_{e}2$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.