Explanation

(a)
We have, ${(1 + x)}^{n} = 1 + nx + \frac{n(n - 1)}{2!}x^{2} + ...$
If $x$ is replace by $–\left( 1 - \frac{1}{x} \right)$ and $n$ is $–n$, then expression
becomes $\left\lbrack 1 - \left( 1 - \frac{1}{x} \right) \right\rbrack^{- n}$
$= 1 + ( - n)\left\lbrack - \left( 1 - \frac{1}{x} \right) \right\rbrack + \frac{( - n)( - n - 1)}{2!}\left\lbrack - \left( 1 - \frac{1}{x} \right) \right\rbrack^{2} + ...$
$\Longrightarrow x^{n} = 1 + n\left( 1 - \frac{1}{x} \right) + \frac{n(n + 1)}{2!}\left( 1 - \frac{1}{x} \right)^{2} + ...$