Binomial Theorem

Theorem

Integral Index

Let $X$ be one of the standard number systems $\N$, $\Z$, $\Q$, $\R$ or $\C$.

Let $x, y \in X$.

Then:

$\ds \forall n \in \Z_{\ge 0}: \, $

$\ds \paren {x + y}^n$

$=$

$\ds \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$

$\ds $

$=$

$\ds x^n + \binom n 1 x^{n - 1} y + \binom n 2 x^{n - 2} y^2 + \binom n 3 x^{n - 3} y^3 + \cdots$

$\ds $

$=$

$\ds x^n + n x^{n - 1} y + \frac {n \paren {n - 1} } {2!} x^{n - 2} y^2 + \frac {n \paren {n - 1} \paren {n - 3} } {3!} x^{n - 3} y^3 + \cdots$

where $\dbinom n k$ is $n$ choose $k$.

Ring Theory

Let $\struct {R, +, \odot}$ be a ringoid such that $\struct {R, \odot}$ is a commutative semigroup.

Let $n \in \Z: n \ge 2$.

Then:

$\ds \forall x, y \in R: \odot^n \paren {x + y} = \odot^n x + \sum_{k \mathop = 1}^{n - 1} \binom n k \paren {\odot^{n - k} x} \odot \paren {\odot^k y} + \odot^n y$

where $\dbinom n k = \dfrac {n!} {k! \ \paren {n - k}!}$ (see Binomial Coefficient).

If $\struct {R, \odot}$ has an identity element $e$, then:

$\ds \forall x, y \in R: \odot^n \paren {x + y} = \sum_{k \mathop = 0}^n \binom n k \paren {\odot^{n - k} x} \odot \paren {\odot^k y}$

General Binomial Theorem

Let $\alpha \in \R$ be a real number.

Let $x \in \R$ be a real number such that $\size x < 1$.

Then:

$\ds \paren {1 + x}^\alpha$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n$

$\ds $

$=$

$\ds 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots$

where:

$\alpha^{\underline n}$ denotes the falling factorial

$\dbinom \alpha n$ denotes a binomial coefficient.

Multiindices

Let $\alpha$ be a multiindex, indexed by $\set {1, \ldots, n}$ such that $\alpha_j \ge 0$ for $j = 1, \ldots, n$.

Let $x = \tuple {x_1, \ldots, x_n}$ and $y = \tuple {y_1, \ldots, y_n}$ be ordered tuples of real numbers.

Then:

$\ds \paren {x + y}^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} \dbinom \alpha \beta x^\beta y^{\alpha - \beta}$

where $\dbinom n k$ is a binomial coefficient.

Extended Binomial Theorem

Let $r, \alpha \in \C$ be complex numbers.

Let $z \in \C$ be a complex number such that $\cmod z < 1$.

Then:

$\ds \paren {1 + z}^r = \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} z^{\alpha + k}$

where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.

Abel's Generalisation

$\ds \paren {x + y}^n = \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}$

$\ds \paren {x + y}^n = \sum x \paren {x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}^{\epsilon_1 + \cdots + \epsilon_n - 1} \paren {y - \epsilon_1 z_1 - \cdots - \epsilon_n z_n}^{n - \epsilon_1 - \cdots - \epsilon_n}$

where the summation ranges over all $2^n$ choices of $\epsilon_1, \ldots, \epsilon_n = 0$ or $1$ independently.

Approximations

Consider the General Binomial Theorem:

$\paren {1 + x}^\alpha = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots$

When $x$ is small it is often possible to neglect terms in $x$ higher than a certain power of $x$, and use what is left as an approximation to $\paren {1 + x}^\alpha$.

This article is complete as far as it goes, but it could do with expansion.
In particular: Put the above into more rigorous language and move the above into a "motivation" section
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First Order

When $x$ is sufficiently small that $x^2$ can be neglected then:

$\paren {1 + x}^\alpha \approx 1 + \alpha x$

and the error is of the order of $\dfrac {\alpha \paren {\alpha - 1} } 2 x^2$

Second Order

When $x$ is sufficiently small that $x^3$ can be neglected then:

$\paren {1 + x}^\alpha \approx 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } 2 x^2$

and the error is of the order of $\dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 3} } 6 x^3$

Also known as

This result is also known as the binomial formula.

Also see

Binomial Theorem

Contents

Theorem

Integral Index

Ring Theory

General Binomial Theorem

Multiindices

Extended Binomial Theorem

Abel's Generalisation

Hurwitz's Generalisation

Approximations

First Order

Second Order

Examples

Cube of Sum

Cube of Difference

Fourth Power of Sum

Fourth Power of Difference

Fifth Power of Sum

Fifth Power of Difference

Sixth Power of Sum

Sixth Power of Difference

Power of $11$: $11^4$