A) \[{{\log }_{e}}\left( \frac{4}{e} \right)\]
B) \[\frac{4}{e}\]
C) \[{{\log }_{e}}\left( \frac{e}{4} \right)\]
D) \[\frac{e}{4}\]
Correct Answer: B
Solution :
\[S=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\frac{1}{4.5}+.....\infty \] \[=1-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}+\frac{1}{5}+.....\infty \] \[=1+2\left[ -\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+.....\infty \right]\] \[=1+2[{{\log }_{e}}2-1]=1+2{{\log }_{e}}2-2\] \[={{\log }_{e}}4-1={{\log }_{e}}4-{{\log }_{e}}e\] \[={{\log }_{e}}\left( \frac{4}{e} \right)\]; \[\therefore \,\,{{e}^{S}}=\frac{4}{e}\].
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