A) \[{{\log }_{e}}2-{{\log }_{e}}3\]
B) \[{{\log }_{e}}3-{{\log }_{e}}2\]
C) \[{{\log }_{e}}6\]
D) None of these
Correct Answer: B
Solution :
\[\frac{1}{3}+\frac{1}{{{2.3}^{2}}}+\frac{1}{{{3.3}^{3}}}+\frac{1}{{{4.3}^{4}}}+......\infty \] \[=\left( \frac{1}{3} \right)+\frac{{{(1/3)}^{2}}}{2}+\frac{{{(1/3)}^{3}}}{3}+\frac{{{(1/3)}^{4}}}{4}+......\infty \] \[=-{{\log }_{e}}\left( 1-\frac{1}{3} \right)=-{{\log }_{e}}\left( \frac{2}{3} \right)={{\log }_{e}}\left( \frac{3}{2} \right)\] \[={{\log }_{e}}3-{{\log }_{e}}2\].
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