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