A) \[\frac{1}{r\,{{\log }_{e}}10}\]
B) \[-\frac{1}{r{{\log }_{e}}10}\]
C) \[-\frac{1}{r!{{\log }_{e}}10}\]
D) None of these
Correct Answer: A
Solution :
We have \[{{\log }_{10}}\left( \frac{n}{n-1} \right)={{\log }_{e}}\left( \frac{n}{n-1} \right).{{\log }_{10}}e\] \[=-{{\log }_{e}}\left( \frac{n-1}{n} \right).{{\log }_{10}}e\]\[=-{{\log }_{10}}e.{{\log }_{e}}\left( 1-\frac{1}{n} \right)\] Therefore coefficient of \[{{n}^{-r}}\]is \[\frac{1}{r}{{\log }_{10}}e=\frac{1}{r{{\log }_{e}}10}\].
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