A formula first derived by Newton, giving (1+z) a, the result of raising 1 + z to an arbitrary power a, as a sequence of form (1+z) a = 1 + A1z + A2z 2 + A3z 3 + .... where the terms Ai (i = 1,2,3...) are given by the formula and where a can be positive, negative, fractional or whole. When the magnitude of z is less than 1, the higher powers get smaller and smaller and the formula can be made as precise as one wishes by including enough of them (for z of small magnitude, 1-2 terms are sufficient), although the result is never exact. For magnitudes of z equal to 1 or more, the formula only holds for values of a which are positive whole numbers. In that case, for any z, the result is exact and the sum of terms with powers of z does not go on arbitrarily but ends with z a.

In mathematics, a theorem that specifies the complete expansion of a binomial raised to any positive integer power.

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