A) \[2\,{{\log }_{e}}2-1\]
B) \[{{\log }_{e}}2-1\]
C) \[{{\log }_{e}}2\]
D) None of these
Correct Answer: A
Solution :
\[S=\frac{1}{2}\left( \frac{1}{1}-\frac{1}{3} \right)+\frac{1}{{{2}^{2}}}\left( \frac{1}{3}-\frac{1}{5} \right)+....\] \[=\left( \frac{1}{1.2}-\frac{1}{2.3} \right)+\left( \frac{1}{3.4}-\frac{1}{4.5} \right)+....=2{{\log }_{e}}2-1\] .
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