Binomial Series
(1 + x)^{\alpha} = 1 + \alpha\, x + \cfrac{1}{2!}\, \alpha\, (\alpha - 1)\, x^2 + \cfrac{1}{3!}\, \alpha\, (\alpha - 1)\, (\alpha - 2)\, x^3 + \ … \ + {\alpha \choose l}\, x^k + \ …$
${\alpha \choose k} = \cfrac{\alpha\, (\alpha - 1) \ … \ (\alpha - k + 1)}{k!}$
Examples:
- $\alpha = 2: 1 + 2x + x^2$
- $\alpha = -1: (1 - x + x^2 - x^3 + x^4 - \ …)$
-
$\alpha = \cfrac{1}{2}: 1 + \cfrac{1}{2}\, x - \cfrac{1}{8}\, x^2 + \cfrac{1}{16}\, x^3 - \ …$ (for $ x <1$)