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