Binomial Series

Summary

The binomial series expands $(1 + x)^{α}$ for real (or complex) $α$ as a power series valid for $| x | < 1$ . Coefficients use falling products, not ordinary factorials, when $α$ is not a nonnegative integer.

Prerequisites

Maclaurin Series, Power Series

Main Result / Formula

(1 + x)^{α} = \sum_{k = 0}^{\infty} (\binom{α}{k}) x^{k}, | x | < 1,

where

(\binom{α}{k}) = \frac{α (α - 1) \dots (α - k + 1)}{k!} (k \geq 1), (\binom{α}{0}) = 1.

If $α$ is a nonnegative integer, the series terminates and becomes the ordinary binomial theorem for all $x$ .

Worked Example

(1 + x)^{- 1 / 2} = 1 - \frac{1}{2} x + \frac{(- 1 / 2) (- 3 / 2)}{2!} x^{2} + \dots, | x | < 1.

Common Mistakes

Writing $(\binom{α}{k}) = α! / (k! (α - k)!)$ for non-integer $α$ .

Connections

Related: Taylor Series, Power Series

References

The generalized binomial series is standard advanced calculus material.^[1]

OpenStax, Calculus Volume 2, https://openstax.org/details/books/calculus-volume-2 ↩︎