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

\begin{align} & 1+ \left(\frac{1}{2}\right)x + \frac{\frac{1}{2}\times \left(\frac{1}{2}-1 \right)}{2!}x^2 + \frac{\frac{1}{2}\times \left(\frac{1}{2}-1 \right) \times \left(\frac{1}{2}-2 \right) }{3!}x^3 + ... \\ = & \, 1+\frac{x}{2}- \frac{x^2}{8}+ \frac{3x^3}{48} +... \\ \\ Range\ & of\ validity\,, \quad|x|\lt1 \end{align}

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

\begin{align} & 1+ \left(\frac{1}{2}\right)(-3x) + \frac{\frac{1}{2}\times \left(\frac{1}{2}-1 \right)}{2!}(-3x)^2 + \frac{\frac{1}{2}\times \left(\frac{1}{2}-1 \right) \times \left(\frac{1}{2}-2 \right) }{3!}(-3x)^3 + ... \\ = & \, 1-\frac{3x}{2}- \frac{9x^2}{8}- \frac{81x^3}{48} +... \\ \\ Range\ & of\ validity\,, \quad|x|\lt \frac{1}{3} \end{align}