Question

The coefficient of $x^n$ in the expansion of ${\log }_e\left(\frac{1}{1+x+x^2+x^3}\right)$, when $n$ is odd is equal to

• A. $-n$
• B. $-\frac{1}{n}$
• C. $\frac{1}{n}$
• D. $n$

Answer 1

The correct option is B $-\frac{1}{n}$

${\log }_e\left(\frac{1}{1+x+x^2+x^3}\right)=-{\log }_e(1+x+x^2+x^3)=-{\log }_e\left[(1+x)(1+x^2)\right]=-{\log }_e(1+x)-{\log }_e(1+x^2)=-(x-\frac{x^2}{2}+\frac{x^3}{3}-.....+\frac{x^n}{n}+...)-(x^2-\frac{x^4}{2}+\frac{x^6}{3}+....)$
It is given that $n$ is odd. Hence, the second part of the above expression will not have $x^n$ term.
Hence, the coefficient of $x^n$ is $-\frac{1}{n}$