Q. The coefficients of $x^{n}$ in the expansion of $\log_{a}{(1 + x)}$ is
A) $\frac{( - 1)^{n - 1}}{n}$
B) $\frac{( - 1)^{n - 1}}{n}\log_{a}e$
C) $\frac{( - 1)^{n - 1}}{n}\log_{e}a$
D) $\frac{( - 1)^{n}}{n}\log_{a}e$
Explanation
(b)
$\log_{a}{(1 + x)} = \log_{e}{(1 + x)}\log_{a}e = \left( \log_{a}e \right)\left\lbrack \sum_{n = 1}^{\infty}{( - 1)^{n - 1}\frac{x^{n}}{n}} \right\rbrack$
So, the coefficient of $x^{n}$ in $\log_{a}{(1 + x)}\$ is $\frac{( - 1)^{n - 1}}{n}\log_{a}e$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.