Q. The coefficient of $x^{n}$ in the series $1 + \frac{a + bx}{1!} + \frac{(a + bx)^{2}}{2!} + \frac{(a + bx)^{3}}{3!} = \ldots\infty$ is
A) $\frac{(ab)^{n}}{n!}$
B) $e^{b}.\frac{a^{n}}{n!}$
C) $e^{a}.\frac{b^{n}}{n!}$
D) $e^{a + b}.\frac{(ab)^{n}}{n!}$
Explanation
(c)
$1 + \frac{(a + bx) =}{1!} + \frac{(a + bx)^{2}}{2!} + \frac{(a + bx)^{3}}{3!} = \ldots\infty = e^{a + bx}$
$\therefore$ Coefficient of$\ x^{n}in\ e^{a}e^{bx} = e^{a}.\frac{(b)^{n}}{n!}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.