Find the first four terms of the expansion of \(( 1+x)^{-4} \) and state the range of validity.

\begin{align} & 1+ (-4)x + \frac{(-4)\times (-4 - 1) }{2!}x^2 + \frac{(-4)\times (-4-1) \times (-4-2) }{3!}x^3 + ... \\ \\ = & \, 1-4x + 10x^2 - 20x^3 +... \\ \\ Range\ & of\ validity\,, \quad|x|\lt1 \end{align}

Find the first four terms of the expansion of \((1-2x)^{-4 }\) and state the range of validity

\begin{align} & 1+ (-4)(-2x) + \frac{(-4)\times (-4 - 1) }{2!}(-2x)^2 + \frac{(-4)\times (-4-1) \times (-4-2) }{3!}(-2x)^3 + ... \\ \\ = & \, 1+8x + 40x^2 + 160x^3 +... \\ \\ Range\ & of\ validity\,, \quad|x|\lt\frac{1}{2} \end{align}