The
binomial theorem is an important formula about the expansion of powers of sums. Its simplest version reads
- <math>(x+y)^n=\sum_{k=0}^n{n \choose k}x^ky^{n-k}\quad\quad\quad(1)</math>
whenever
n is any non-negative integer and the numbers
- <math>{n \choose k}=\frac{n!}{k!(n-k)!}</math>
are the
binomial coefficients. This formula, and the
triangular arrangement of the binomial coefficients, are often attributed to
Blaise Pascal who described them in the
17th century. It was however known long before to Chinese mathematicians.
The cases n=2, n=3 and n=4 are the ones most commonly used:
- (x + y)^{2} = x^{2} + 2xy + y^{2}
- (x + y)^{3} = x^{3} + 3x^{2}y + 3xy^{2} + y^{3}
- (x + y)^{4} = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}
Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a ring as long as xy = yx.
Isaac Newton generalized the formula to other exponents by considering an infinite series:
- <math>(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^k y^{r-k}\quad\quad\quad(2)</math>
where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by
- <math>{r \choose k}=\frac{r(r-1)(r-2)\cdots(r-k+1)}{k!}</math>
(which in case
k = 0 is a
product of no numbers at all and therefore equal to 1, and in case
k = 1 is equal to
r, as the additional factors (
r - 1), etc., do not appear in that case).
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
- <math>\left\{\,x^k:k=0,1,2,\dots\,\right\}</math>
is of
binomial type.
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