Q. The value of $\frac{1}{2!} + \frac{2}{3!} + ... + \frac{999}{1000!}$ is equal to
A) $\frac{1000! - 1}{1000!}$
B) $\frac{1000! + 1}{1000!}$
C) $\frac{999! - 1}{999!}$
D) $\frac{999! + 1}{999!}$
Explanation
(a)
$\frac{1}{2!} + \frac{2}{3!} + ... + \frac{999}{1000!}$
$= \frac{2 - 1}{2!} + \frac{3 - 1}{3!} + ... + \frac{1000 - 1}{1000!}$
$= \frac{1}{1!} - \frac{1}{2!} + \frac{1}{2!} - \frac{1}{3!} + ... + \frac{1}{999!} - \frac{1}{1000!}$
$1 - \frac{1}{1000!} = \frac{1000! - 1}{1000!}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.