Binomial theorem - Aetherwisp

The binomial theorem, or binomial expansion, describes the algebraic expansion of powers in a binomial $(x+y)^{n}$ . $n$ is typically an integer, but there are generalization for real and complex numbers as well, which are particularly useful to approximate square roots.

> where the **binomial coefficient** is defined as > $$\begin{pmatrix}n \\ k\end{pmatrix}=\frac{n!}{k!(n-k)!}

Newton also devised a generalization that allows the binomial theorem to apply to real and complex exponents too. Here, the finite sum becomes as series.

> where the binomial coefficient is defined as > $$\begin{pmatrix}r \\ k\end{pmatrix}=\frac{(r)_{k}}{k!}

with $(r)_{k}=r(r-1)\ldots(r-k+1)$ the falling factorial function.

This generalization gives rise to useful approximations for the square root and its inverse, where $r=1/2$ and $r=-1/2$ respectively:

\sqrt{ 1+x }=1+ \frac{1}{2}x- \frac{1}{8}x^{2}+ \frac{1}{16}x^{3} - \frac{5}{128}x^{4}+\ldots

\frac{1}{\sqrt{ 1+x }}=1- \frac{1}{2}x+ \frac{3}{8}x^{2}- \frac{5}{8}x^{3}+ \frac{35}{128}x^{4}+\ldots

\frac{1}{\sqrt{ 1-x }}=1+ \frac{1}{2}x+ \frac{3}{8}x^{2}+ \frac{5}{8}x^{3}+ \frac{35}{128}x^{4}+\ldots

These approximations are best when $x$ is very small.