The binomial theorem is an important formula about the expansion of powers of sums. Its simplest version reads
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
Isaac Newton generalized the formula to other exponents by considering an infinite 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.
The binomial theorem can be stated by saying that the polynomial sequence