A) \[{{\log }_{e}}\sqrt{\frac{3}{2}}\]
B) \[{{\log }_{e}}\sqrt{3}\]
C) \[{{\log }_{e}}\sqrt{\frac{7}{2}}\]
D) \[{{\log }_{e}}3\]
Correct Answer: B
Solution :
Let \[S=\frac{1}{2}+\frac{1}{3}\cdot \frac{1}{{{2}^{3}}}+\frac{1}{5}\cdot \frac{1}{{{2}^{5}}}+...\] \[=\frac{1}{2}\log \left( \frac{1+\frac{1}{2}}{1-\frac{1}{2}} \right)\] \[\left[ \because \,\,\log \left( \frac{1+x}{1-x} \right)=2\left( x+\frac{{{x}^{3}}}{3}+... \right) \right]\] \[=\frac{1}{2}\log \left( \frac{3}{1} \right)=\log {{3}^{1/2}}\] \[=\log \sqrt{3}\]
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