A) \[{{\log }_{e}}\frac{1}{2}\]
B) \[2\,\left[ x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+.....\infty \right]\]
C) \[2\,\left[ {{x}^{2}}+\frac{{{x}^{4}}}{4}+\frac{{{x}^{6}}}{6}+.....\infty \right]\]
D) None of these
Correct Answer: A
Solution :
\[{{\log }_{e}}\sqrt{\frac{1+x}{1-x}}=\frac{1}{2}{{\log }_{e}}\frac{1+x}{1-x}\] \[=\frac{1}{2}.2\left\{ x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+...... \right\}=x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+......\]
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