Binomial theorem

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In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads

$(x+y)^n=\sum_{k=0}^n{n \choose k}x^ky^{n-k}\quad\quad\quad(1)$

whenever n is any non-negative integer, the numbers

${n \choose k}=\frac{n!}{k!\,(n-k)!}$

are the binomial coefficients, and $n!$ denotes the factorial of n.

This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar Khayyám in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.

For example, here are the cases n = 2, n = 3 and n = 4:

$(x + y)^2 = x^2 + 2xy + y^2\,$

$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\,$

$(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.\,$

Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a semiring as long as xy = yx.

Newton's generalized binomial theorem

Isaac Newton generalized the formula to other exponents by considering an infinite series:

${(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^k y^{r-k}\quad\quad\quad(2)}$

where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by

${r \choose k}={1 \over k!}\prod_{n=0}^{k-1}(r-n)=\frac{r(r-1)(r-2)\cdots(r-(k-1))}{k!}\,$

In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r − 1), etc., do not appear.

Another way to express this quantity is

${r \choose k}=\frac{(-1)^k}{k!}(-r)_k,$

which is important when one is working with infinite series and would like to represent them in terms of generalized hypergeometric functions. The notation $(\cdot)_k$ is the Pochhammer symbol. This form is vital in applied mathematics, for example, when evaluating the formulas that model the statistical properties of the phase-front curvature of a light wave as it propagates through optical atmospheric turbulence.

A particularly handy but non-obvious form holds for the reciprocal power:

$\frac{1}{(1-x)^r}=\sum_{k=0}^\infty {r+k-1 \choose k} x^k \equiv \sum_{k=0}^\infty {r+k-1 \choose r-1} x^k.$

For a more extensive account of Newton's generalized binomial theorem, see binomial series.

The sum in (2) converges and the equality is true whenever the real or complex numbers x and y are "close together" in the sense that the absolute value | x/y | is less than one.

The geometric series is a special case of (2) where we choose y = 1 and r = −1.

Formula (2) is also valid for elements x and y of a Banach algebra as long as xy = yx, y is invertible and ||x/y|| < 1.

"Binomial type"

The binomial theorem can be stated by saying that the polynomial sequence

$\left\{\,x^k:k=0,1,2,\dots\,\right\}\,$

is of binomial type.

Proof

One way to prove the binomial theorem is with mathematical induction. When n = 0, we have

$(a+b)^0 = 1 = \sum_{k=0}^0 { 0 \choose k } a^{0-k}b^k.$

For the inductive step, assume the theorem holds when the exponent is $m$. Then for n = m + 1

$(a+b)^{m+1} = a(a+b)^m + b(a+b)^m \,$

$= a \sum_{k=0}^m { m \choose k } a^{m-k} b^k + b \sum_{j=0}^m { m \choose j } a^{m-j} b^j$ by the inductive hypothesis

$= \sum_{k=0}^m { m \choose k } a^{m-k+1} b^k + \sum_{j=0}^m { m \choose j } a^{m-j} b^{j+1}$ by multiplying through by $a$ and $b$

$= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1} b^k + \sum_{j=0}^m { m \choose j } a^{m-j} b^{j+1}$ by pulling out the $k=0$ term

$= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1} b^k + \sum_{k=1}^{m+1} { m \choose k-1 }a^{m-k+1}b^{k}$ by letting $j = k-1$

$= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1}b^k + \sum_{k=1}^{m} { m \choose k-1 }a^{m+1-k}b^{k} + b^{m+1}$ by pulling out the $k=m+1$ term from the RHS

$= a^{m+1} + b^{m+1} + \sum_{k=1}^m \left[ { m \choose k } + { m \choose k-1 } \right] a^{m+1-k}b^k$ by combining the sums

$= a^{m+1} + b^{m+1} + \sum_{k=1}^m { m+1 \choose k } a^{m+1-k}b^k$ from Pascal's rule

$= \sum_{k=0}^{m+1} { m+1 \choose k } a^{m+1-k}b^k$ by adding in the $m+1$ terms.

as desired.

Trivia

• In the Sherlock Holmes books, the villain Professor Moriarty is the author of A Treatise on the Binomial Theorem.

• The binomial theorem is mentioned in the Gilbert and Sullivan song I am the Very Model of a Modern Major General.

• The binomial theorem appears in at least three different works by Monty Python - Coal Mine in Llandarogh Carmarthen, The Tale of Happy Valley, and The Meaning of Life.

• The binomial theorem is mentioned in the TV series NUMB3RS in episode #217 ("Mind Games") in Season 2.

See also

• Multinomial theorem
• Pascal's triangle

This article incorporates material from inductive proof of binomial theorem on PlanetMath, which is licensed under the GFDL.

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