3.6 The general binomial series
In the case where , the expansion of is given by
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
This expression can be expanded, but only after some rearranging. First, the expression can be rewritten using indices to give
This is almost, but not quite, in the form - the four needs to be taken out first to give
Applying the fact that
This can then be expanded using the general binomial series.