Q. The value of $\sum_{n = 0}^{\infty}\left( \frac{2i}{3} \right)^{n}$ is
A) $\frac{9 + 6i}{13}$
B) $\frac{9 - 6i}{13}$
C) $9 + 6i$
D) $9 - 6i$
Explanation
(a)
$\sum_{n = 0}^{\infty}{\left( \frac{2i}{3} \right)^{n} = 1 + \left( \frac{2i}{3} \right) + \left( \frac{2i}{3} \right)^{2} + \left( \frac{2i}{3} \right)^{3} + ...}$
$= \frac{1}{1 - \frac{2i}{3}} = \frac{3}{3 - 2i} \times \frac{3 + 2i}{3 + 2i}$
$= \frac{9 + 6i}{13}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.