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