We have several closely related results that are variously known as the binomial theorem according to the source. More confusing is the fact that some of these (and others) closely related results are variously known as the binomial formula, the binomial expansion and binomial identity, with identity itself sometimes simply called “binomial series” rather than “binomial theorem”.

A more general case of the binomial theorem is the identity of the binomial series

 (x+a)^nu=sum_(k=0)^infty(nu; k)x^ka^(nu-k),

where (nu; k) is a binomial coefficient and nu is a real number. That series converges for nu>=0 an integer, or |x/a|<1. The general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with a=1 the binomial theorem.

When nu is a positive integer n, it ends with n=nu and can be written in the form

 (x+a)^n=sum_(k=0)^n(n; k)x^ka^(n-k).

That form of identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).

The various terminologies are summarized in the following table.

“binomial theorem”source
Graham et al. (1994, p. 162)
Arfken (1985, p. 307)
Abramowitz and Stegun (1972, p. 10)

“source” binomial theorem

Abramowitz and Stegun (1972, p. 10)

This binomial theorem was known for the case n=2 of Euclid around 300 BC, and declared in its modern form by Pascal in a posthumous pamphlet published in 1665. The pamphlet by Pascal, together with the correspondence on the topic with Fermat since 1654 (and published in 1679) is the basis for naming the arithmetic triangle in his honour.

The formula was also shown by Newton (1676) for negative integers -n,

 (x+a)^(-n)=sum_(k=0)^infty(-n; k)x^ka^(-n-k),

which is the so-called negative binomial series and converges for |x|<a.

in fact, the generalization

 (1+z)^a=sum_(k=0)^infty(a; k)z^k

holds for all complex z with |z|<1.

His many other talents include Major General Stanley in Gilbert and Sullivan’s operetta “The Pirates of Penzance” which impresses the pirates with his knowledge about the binomial theorem in “The Song of the Major General” as follows: “I have plant, animal and mineral information, I understand the kings of England, and I quote the historical battles, From Marathon to Waterloo, in categorical order; I also know very well mathematical matters, I understand equations, both simple and square, concerning the binomial theorem teeming with news, with many cheerful facts about the square of the hypotenuse”.