A calculus of the absurd

3.6 The general binomial series

In the case where \(\abs {x} < 1\), the expansion of \((1+x)^r\) is given by

\begin{equation} (1 + x)^{r} = 1 + rx + \frac {r(r-1)}{2!} x^2 + \frac {r(r-1)(r-2)}{3!} + ... \end{equation}

This can be proved using Maclaurin series (in the "differential calculus" section).

There are some binomial expansions which look like they can’t be expanded using this formula, but can. For example

\begin{equation} \frac {1}{\sqrt {4-x}} \end{equation}

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

\begin{equation} (4-x)^{-\frac {1}{2}} \end{equation}

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

\begin{equation} \left (4\left (1-\frac {x}{4}\right )\right )^{-\frac {1}{2}} \end{equation}

and then

\begin{equation} 4^{-\frac {1}{2}} \left (1 + \left (-\frac {x}{4}\right )\right )^{-\frac {1}{2}} \end{equation}

Which can then be expanded.