A) \[2\,\,{{\log }_{e}}2\]
B) \[{{\log }_{e}}\,\,2-1\]
C) \[{{\log }_{e}}\,\,2\]
D) \[{{\log }_{e}}\,\,\left( \frac{4}{e} \right)\]
Correct Answer: D
Solution :
Now, \[\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-....\] \[=\left( 1-\frac{1}{2} \right)-\left( \frac{1}{2}-\frac{1}{3} \right)+\left( \frac{1}{3}-\frac{1}{4} \right)-.....\] \[=1-2\,.\,\frac{1}{2}+2\,.\,\frac{1}{3}-2\,.\,\frac{1}{4}+....\] \[=2\,\left( 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.... \right)-1\] \[=2\log (1+1)-1=2\,\log 2-\log e\] \[=\log 4-\log e=\log \frac{4}{e}\]
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