Fractional and negative powers

For a rational exponent \(\alpha\) that is not a non-negative integer, the binomial expansion is an infinite series.

\((1+u)^\alpha=1+\alpha u+\dfrac{\alpha(\alpha-1)}{2!}u^2+\dfrac{\alpha(\alpha-1)(\alpha-2)}{3!}u^3+\cdots\), for \(|u|<1\).

\(\binom\alpha0=1,\qquad\binom\alpha r=\dfrac{\alpha(\alpha-1)\cdots(\alpha-r+1)}{r!}\quad(r\ge1)\). Use this definition, not factorials of fractional or negative numbers.

For a positive integer \(m\), \((1-u)^{-m}=1+mu+\dfrac{m(m+1)}{2!}u^2+\dfrac{m(m+1)(m+2)}{3!}u^3+\cdots\).

The special case \((1-u)^{-1}=1+u+u^2+u^3+\cdots\) is a geometric series.

An infinite series equals the original expression within its validity range. A finite truncation is an approximation. Use \(\approx\) for a truncated polynomial, or retain an ellipsis with an equality.