<sod> The '''binomial theorem''' is an important formula about the expansion of powers of sums. It simplest version reads
:<math>(x+y)^n=\sum_{k=0}^n{n \choose k}x^ky^{n-k}</math>(1)
whenever ''n'' is any positive integer (but see below) and the numbers
 :<math>{n \choose k}=\frac{n!}{k!(n-k)!}</math>
 are the [[binomial coefficient|binomial coefficients]]. This formula, and the [[Pascal's triangle|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.
 
 
 [[Isaac Newton]] generalized the formula to non-integral and negative powers by considering an [[infinite series]]:
 
:<math>(x+y)^r=\sum_{k=0}^\infty C(r, k) x^ky^{r-k}</math> (2)
 
 
 
Here, ''r'' can be any real or complex number, and the sum will converge 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 chose ''y'' = 1 and ''r'' = -1. <eod> <sod>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(1)</math>
whenever ''n'' is any positive integer and the numbers
 :<math>{n \choose k}=\frac{n!}{k!(n-k)!}</math>
 are the [[binomial coefficient|binomial coefficients]]. This formula, and the [[Pascal's triangle|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.
 
 
 [[Isaac Newton]] generalized the formula to non-integral and negative powers by considering an [[infinite series]]:
 
:<math>(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^ky^{r-k}\quad\quad(2)</math> 
 
Here, ''r'' can be any real or complex number, and the sum will converge 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 definition of binomial coefficients for non-integral arguments is given in the [[binomial coefficient]] article.
 
The [[geometric series]] is a special case of (2) where we chose ''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. <eod>