Binomial Theorem

Equivalent forms

(1+x)^\alpha=\sum_{k=0}^{\infty}\binom{\alpha}{k}x^{k}

Pascal's triangle, the normal distribution, and every 'small-x expansion' in physics all fall out of one counting argument.

Symbol	Name	SI unit	Dimension	Range
$n$	Exponent Power to expand (integer or real)	1	1	1 – 10
$k$	Term index Which term in the expansion	1	1	0 – 10
$\binom{n}{k}$	Binomial coefficientoutput Number of ways to choose k from n	1	1	—
$x$	Variable First term of the binomial	1	1	—

Where it holds

Finite sum for non-negative integer n. For real/negative

\alpha the

infinite series

(1+x)^\alpha

converges only for |x|<1 (and at endpoints depending

on \alpha )

.

Dimensional analysis

(x+y)^n

\Sigma C(n

,k) x^{n-k} y^k

If x,y share a dimension D, every term has dimension

D^{n}

; the coefficients are pure numbers.

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Discovery

Isaac Newton (generalized); ancient roots · 1665

The integer case was known to Pascal, al-Karaji, and Chinese mathematician Yang Hui centuries earlier. Newton's 1665 generalization to fractional and negative exponents produced infinite series and was pivotal to his invention of calculus — he could now expand and integrate (1+x)^α term by term.

Research status: stable

Real-world applications

First-order approximations throughout physics (e.g. $\gamma \approx 1+v^{2}/2c^{2})$
Deriving the normal distribution as a limit of the binomial
Series expansions in perturbation theory

Common misconceptions

For non-integer $\alpha the$ series is infinite and needs |x|<1 to converge
C(n,k) counts combinations, not permutations
The approximation $(1+x)^\alpha \approx 1+\alpha x$ only holds for small x

Experimental verification

A mathematical identity; verified by direct expansion and by the convergence of physical approximations (relativistic energy, lens equations) that use its leading terms.

Derivation

Write

(x+y)^{n} as n

copies of (x+y) multiplied together.

Expanding, every term is a product of one choice (x or y) from each factor; a term with k y's and

(n-k) x's

occurs as many times as there are ways to choose which k factors give y, namely C(n,

k)=n!/(k!(n-k)!)

.

Summing over k gives the theorem.

Newton extended it by matching Taylor coefficients, defining

C(\alpha

,

k)=\alpha (\alpha -1)

…

(\alpha -k+1)/k!

for

any \alpha

.

What if…

What if n is a large integer and we

set x=y=1

?

$\Sigma C(n$ , $k) = 2^{n}$ — the total number of subsets, and the binomial coefficients trace out the bell curve as n grows.

What if |x| > 1 in the generalized series?

The infinite series diverges; you must factor out the larger term first to make the remaining ratio small.

1

Relativistic kinetic energy

Given ·

\gamma = (1-\beta ^{2})^{-1/2}

,

\beta =v/c

≪ 1

Find · leading correction

Steps

$Use (1+u)^{-1/2} \approx 1 - \tfrac{1}{2}u$ with $u=-\beta ^{2}$
$\gamma \approx 1 + \tfrac{1}{2}\beta ^{2}$
$(\gamma -1)mc^{2} \approx \tfrac{1}{2}mv^{2}$ recovers Newtonian KE

Result ·

\gamma \approx 1 + \tfrac{1}{2}\beta ^{2}

, so

KE \approx \tfrac{1}{2}mv^{2}

taylor seriesBinomial Distribution→pascals triangle