A) \[{{\log }_{e}}3\]
B) \[{{\log }_{e}}4\]
C) \[{{\log }_{e}}\left( \frac{e}{2} \right)\]
D) \[{{\log }_{e}}\left( \frac{2}{3} \right)\]
Correct Answer: B
Solution :
\[S=1+2\left\{ \frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+.... \right\}\] \[=\left( 2-\frac{2}{2} \right)+2\left\{ \frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+.... \right\}\] \[=2\left\{ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+... \right\}=2{{\log }_{e}}2\] = \[{{\log }_{e}}4\].
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