A) \[{{\log }_{e}}\left( \frac{n+1}{n} \right)\]
B) \[{{\log }_{e}}\left( \frac{n}{n+1} \right)\]
C) \[{{\log }_{e}}\left( \frac{n-1}{n} \right)\]
D) \[{{\log }_{e}}\left( \frac{n}{n-1} \right)\]
Correct Answer: A
Solution :
\[\frac{1}{n}-\frac{1}{2{{n}^{2}}}+\frac{1}{3{{n}^{3}}}-\frac{1}{4{{n}^{4}}}+....\] \[=\frac{1}{n}-\frac{{{(1/n)}^{2}}}{2}+\frac{{{(1/n)}^{3}}}{3}-\frac{{{(1/n)}^{4}}}{4}+....\] \[={{\log }_{e}}\left( 1+\frac{1}{n} \right)={{\log }_{e}}\left( \frac{n+1}{n} \right)\].
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