The binomial expansion we have learned so far will not help us to expand

because $\frac{1}{2}$ is not an integer. Fortunately, the binomial expansion formula can be generalised:

This expansion could go on forever, and so we usually choose some term to stop at and just get an approximation.

The formula looks a bit unpleasant but it isn't so bad once you've seen some examples.

Find the expansion, up to and including the term in $x^3,$ of

1. $$\sqrt{1+x}$$
2. $$\frac{1}{1-2x}$$
3. $$(4+x{)}^{\frac{3}{2}}$$
4. $$\frac{1+x}{2-x}$$

Give the range of validity of the expansion in each case.