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}=*x*^{2}+ 2*xy*+*y*^{2} - (
*x*+*y*)^{3}=*x*^{3}+ 3*x*+ 3^{2}y*xy*+^{2}*y*^{3} - (
*x*+*y*)^{4}=*x*^{4}+ 4*x*+ 6^{3}y*x*+ 4^{2}y^{2}*xy*+^{3}*y*^{4}

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