Binomial series

Example

Let $\alpha$ be a real number. For each natural number $k$, let $$\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) \cdots (\alpha - k + 1)}{k!}.$$ The binomial series is the power series $\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$. For $|x| < 1$ its sum is $(1 + x)^{\alpha}$.

Remarks

If $\alpha$ is a natural number, then $\binom{\alpha}{k} = 0$ for $k > \alpha$. The series is then the binomial formula for $(1 + x)^{\alpha}$. Otherwise the radius of convergence is 1. For $\alpha = -1$ it is the geometric series $\sum (-1)^k x^k = 1/(1 + x)$. For $\alpha = 1/2$ it gives $\sqrt{1 + x} = 1 + x/2 - x^2/8 + \cdots$.

Explanations

L1 human

The binomial series extends the binomial formula to exponents that are not natural numbers. The expansion then has infinitely many terms, and it holds only for small $x$.