Binomial theorem

From the Quicksilver Metaweb.

The binomial theorem is an important formula about 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 and the numbers ${n \choose k}=\frac{n!}{k!(n-k)!}$ are the binomial coefficients. This formula, and the triangular arrangement[1] of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was however known long before to Chinese mathematicians.

The cases n=2, n=3 and n=4 are the ones most commonly used: (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

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

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 ${r \choose k}=\frac{r(r-1)(r-2)\cdots(r-k+1)}{k!}$ (which in case k = 0 is a product of no numbers at all and therefore equal to 1, and in case k = 1 is equal to r, as the additional factors (r - 1), etc., do not appear in that case).

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 of x/y is less than one.

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

footnote 1. pascal's triangle

grabbed/edited from the Wikipedia http://en.wikipedia.org/wiki/Binomial_theorem staying within the Baroque Era.