A) \[\frac{x}{1+x}-{{\log }_{e}}(1-x)\]
B) \[\frac{x}{1+x}+{{\log }_{e}}(1-x)\]
C) \[\frac{x}{1-x}-{{\log }_{e}}(1-x)\]
D) \[\frac{x}{1-x}+{{\log }_{e}}(1-x)\]
Correct Answer: D
Solution :
\[S=\left( 1-\frac{1}{2} \right){{x}^{2}}+\left( 1-\frac{1}{3} \right){{x}^{3}}+\left( 1-\frac{1}{4} \right){{x}^{4}}+............\]\[+\left( 1-\frac{1}{n+1} \right){{x}^{n+1}}+........\infty \] \[=\{{{x}^{2}}+{{x}^{3}}+{{x}^{4}}+.........\}-\left\{ \frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}+.......... \right\}\] \[=\frac{{{x}^{2}}}{1-x}-\{-{{\log }_{e}}(1-x)-x\}=\frac{x}{1-x}+{{\log }_{e}}(1-x)\].
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