Maths Question:
$\begin{align} & \text{Expand }{{(5x-1)}^{-2}} \\ & \text{Giving the first four terms and stating the series is valid} \\\end{align}$
Maths Solution:
$\begin{align} & {{(5x-1)}^{-2}}={{[-1(1-5x)]}^{-2}}={{(1-5x)}^{-2}} \\ & {{(5x-1)}^{-2}}=\left[ 1+(-2)(-5x)+\frac{(-2)(-3)}{2}{{(-5x)}^{2}}+\frac{(-2)(-3)(-4)}{3!}{{(-5x)}^{3}}+--- \right] \\ & {{(5x-1)}^{-2}}=\left[ 1+10x+75{{x}^{2}}+500{{x}^{3}}+---- \right] \\ & \text{The range is valid within }-\frac{1}{5}<x<\frac{1}{5} \\\end{align}$
University mathstopic: