3.6 The general binomial series

In the case where |x|<1\left\lvert x\right\rvert<1, the expansion of (1+x)r(1+x)^{r} is given by

(1+x)r=1+r⁢x+r⁢(r−1)2!⁢x2+r⁢(r−1)⁢(r−2)3!+…(1+x)^{r}=1+rx+\frac{r(r-1)}{2!}x^{2}+\frac{r(r-1)(r-2)}{3!}+... (3.66)

This can be proved using Maclaurin series (in Section 10.12).

There are some binomial expansions where it might not be immediately obvious that they can be expanded using this formula, but can. One example is

14−x\frac{1}{\sqrt{4-x}} (3.67)

This expression can be expanded, but only after some rearranging. First, the expression can be rewritten using indices to give

(4−x)−12(4-x)^{-\frac{1}{2}} (3.68)

This is almost, but not quite, in the form (1+x)(1+x) - the four needs to be taken out first to give

(4⁢(1−x4))−12\left(4\left(1-\frac{x}{4}\right)\right)^{-\frac{1}{2}} (3.69)

Applying the fact that (a⁢b)n=an⁢bn(ab)^{n}=a^{n}b^{n}

4−12⁢(1+(−x4))−124^{-\frac{1}{2}}\left(1+\left(-\frac{x}{4}\right)\right)^{-\frac{1}{2}} (3.70)

This can then be expanded using the general binomial series.