Binomial theorem

In elementary algebra, the binomial theorem is the identity that states that for any non-negative integer n,

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k},$

where

${n \choose k}={\frac {n!}{k!(n-k)!}}.$

One way to prove this identity is by mathematical induction.

Newton's binomial theorem

There is also Newton's binomial theorem, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (x + y)ⁿ as an infinite series when n is not an integer or is not positive.