A) 1/2
B) \[-1\]
C) 1
D) None of these
Correct Answer: A
Solution :
\[2{{\log }_{e}}x-{{\log }_{e}}\left\{ \left( 1+\frac{1}{x} \right)x \right\}-{{\log }_{e}}\left\{ \left( 1-\frac{1}{x} \right)x \right\}\] \[=2{{\log }_{e}}x-\left\{ {{\log }_{e}}\left( 1+\frac{1}{x} \right)+{{\log }_{e}}x \right\}\]\[-\left\{ {{\log }_{e}}\left( 1-\frac{1}{x} \right)+{{\log }_{e}}x \right\}\] \[=-\left\{ {{\log }_{e}}\left( 1+\frac{1}{x} \right)+{{\log }_{e}}\left( 1-\frac{1}{x} \right) \right\}=2\left\{ \frac{1}{2{{x}^{2}}}+\frac{1}{4{{x}^{4}}}+....... \right\}\] The coefficient of\[{{x}^{-4}}=2.\frac{1}{4}=\frac{1}{2}\].
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