A) \[{{x}^{n}}\]
B) \[{{x}^{-n}}\]
C) \[{{\left( 1-\frac{1}{x} \right)}^{n}}\]
D) None of these
Correct Answer: A
Solution :
We have \[{{(1+x)}^{n}}={{\,}^{n}}{{C}_{0}}+{{\,}^{n}}{{C}_{1}}x+{{\,}^{n}}{{C}_{2}}{{x}^{2}}+.....\infty \] If x is replace by \[-\left( 1-\frac{1}{x} \right)\]and n is \[-n\], then expression becomes \[{{\left[ 1-\left( 1-\frac{1}{x} \right) \right]}^{-n}}.\] \[=1+(-n)\,\left[ -\left( 1-\frac{1}{x} \right) \right]+\frac{(-n)(-n-1)}{2!}{{\left[ -\left( 1-\frac{1}{x} \right) \right]}^{2}}+...\] or \[{{x}^{n}}=1+n\left( 1-\frac{1}{x} \right)+\frac{n(n+1)}{2!}{{\left( 1-\frac{1}{x} \right)}^{2}}+....\]
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