Binomial Series

The binomial series extends the familiar binomial theorem from finite polynomial expansions to infinite series expansions. This distinction is important: the binomial theorem applies to positive integer exponents, while the binomial series applies more broadly, including fractional and negative exponents. It is obtained from the Maclaurin series of (1 + x)^m, where m is any real exponent, and the expansion converges for |x| < 1.

The familiar binomial theorem is

\begin{equation*}(y+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k y^{n-k}\end{equation*}

Setting y = 1 gives

\begin{equation*}(1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k\end{equation*}

This idea is generalized by considering f(x) = (1 + x)^m. Using the Maclaurin series,

\begin{equation*}(1+x)^m=1+m x+\jfrac{m(m-1)}{2!}x^2+\frac{m(m-1)(m-2)}{3!}x^3+\cdots\end{equation*}

In compact form,

\begin{equation*}(1+x)^m=\sum_{k=0}^{\infty}\binom{m}{k}x^k\end{equation*}

where

\begin{equation*}\binom{m}{k}= \frac{m(m-1)(m-2)\cdots(m-k+1)}{k!}\end{equation*}

For a positive integer m, the factors eventually include zero, so the series terminates and becomes the binomial theorem. For negative or noninteger m, the factors do not terminate, giving an infinite binomial series.