A) \[2\,\left[ x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+......\infty \right]\]
B) \[\,\left[ x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+......\infty \right]\]
C) \[2\,\left[ \frac{1}{x}+\frac{1}{3{{x}^{3}}}+\frac{1}{5{{x}^{5}}}+...\infty \right]\]
D) \[\,\left[ \frac{1}{x}+\frac{1}{3{{x}^{3}}}+\frac{1}{5{{x}^{5}}}+...\infty \right]\]
Correct Answer: C
Solution :
\[{{\log }_{e}}(x+1)-{{\log }_{e}}(x-1)={{\log }_{e}}\frac{x+1}{x-1}\] \[={{\log }_{e}}\left( \frac{1+\frac{1}{x}}{1-\frac{1}{x}} \right)=2\left\{ \frac{1}{x}+\frac{1}{3{{x}^{3}}}+\frac{1}{5{{x}^{5}}}+....... \right\}\].
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