A) \[{{\log }_{e}}\left( \frac{{{n}^{2}}}{{{n}^{2}}+1} \right)\]
B) \[{{\log }_{e}}\left( \frac{{{n}^{2}}+1}{{{n}^{2}}} \right)\]
C) \[{{\log }_{e}}\left( \frac{{{n}^{2}}}{{{n}^{2}}-1} \right)\]
D) None of these
Correct Answer: C
Solution :
\[S=\frac{\frac{1}{{{n}^{2}}}}{1}+\frac{{{\left( \frac{1}{{{n}^{2}}} \right)}^{2}}}{2}+\frac{{{\left( \frac{1}{{{n}^{2}}} \right)}^{3}}}{3}+....\] \[=-{{\log }_{e}}\left( 1-\frac{1}{{{n}^{2}}} \right)={{\log }_{e}}\frac{{{n}^{2}}}{{{n}^{2}}-1}\]. Trick: Putting\[n=2,\]the sum of the series upto 4 terms is \[\frac{1}{4}+\frac{1}{32}+\frac{1}{192}+\frac{1}{1024}=0.2874\]... and option = - 0.223...., =0.223....., =0.2876..... Hence answer is (c).
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