In mathematics, the binomial series generalizes the purely algebraic binomial theorem. It is a special case of a Newton series. The binomial series is the series Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... In mathematics, a difference operator maps a function, f(x), to another function, f(x + a) − f(x + b). ... In mathematics, a series is often represented as the sum of a sequence of terms. ...

in which

where is the Pochhammer symbol, and in particular In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used...

because it is the empty product. In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...

Note: We do not define to be because we do not assume that α is a positive integer.

The results concerning convergence of this series were discovered by Sir Isaac Newton, and therefore one sometimes speaks of Newton's binomial theorem. Sir Isaac Newton, FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, alchemist, and natural philosopher, regarded by many as the greatest figure in the history of science. ...

Whether the series converges depends on the values of α and x.

• If |x| < 1, the series converges to (1 + x)^α for all α in the real numbers.

• If x = 1, the series converges to 2^α for α > −1.

• If x = −1, the series converges to 0 for α ≥ 0.

In expositions on calculus, the series is typically constructed by formally deriving a power series for (1 + x)^α, and then proving that the power series converges for some x, namely −1 < x < 1 in this case. Convergence can be proved by the ratio test. In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ...

The binomial series generalizes the binomial theorem to noninteger values of α. If α is an integer, then the (α + 1)th term and all later terms in the series are zero, since each one contains a factor equal to (α − α). In that case the summation reduces to the binomial formula. In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

See also

• Table of Newtonian series