Find the first four terms of the expansion of \(( 3+x)^{\frac{1}{2}} \) and state the range of validity.

\begin{align} & ( 3+x)^{\frac{1}{2}} = \left[3^{\frac{1}{2}}\left(1+ \frac{1}{3}x \right)^{\frac{1}{2}} \right] \\ \\ 3^{\frac{1}{2}} & \left( 1+ \left(\frac{1}{3} \right)x + \frac{(\frac{1}{2})\times (\frac{1}{2} - 1) }{2!}\left(\frac{1}{3}x \right)^2 + \frac{\frac{1}{2}\times \left(\frac{1}{2}-1 \right) \times \left(\frac{1}{2}-2 \right) }{3!} \left(\frac{1}{3}x \right)^3 +...\right) \\ \\ = & \, \sqrt{3}+ \frac{\sqrt{3}}{3}x - \frac{\sqrt{3}}{72}x^2 + \frac{\sqrt{3}}{432}x^3 +... \\ \\ Range\ & of\ validity\,, \quad|x|\lt3 \end{align}

Find the first four terms of the expansion of \((4-3x)^{-2}\) and state the range of validity

\begin{align} & ( 4-3x)^{-2} = \left[4^{-2}\left(1- \frac{3}{4}x \right)^{-2} \right] \\ \\ = \, 4^{-2} & \left( 1+ \left(-2 \right) \left(-\frac{3}{4}x\right) + \frac{-2\times (-2 - 1) }{2!}\left(-\frac{3}{4}x \right)^2 + \frac{-2\times \left(-2-1 \right) \times \left(-2-2 \right) }{3!} \left(-\frac{3}{4}x \right)^3 +...\right) \\ \\ = & \, \frac{1}{16}+ \frac{{6}}{64}x + \frac{27}{256}x^2 + \frac{108}{1024}x^3 +... \\ \\ & Range\ of\ validity\,, \quad|x|\lt \frac{4}{3} \end{align}