For the binomial ((1+x)^n), where (n) is any brational number/b, e.g. fractions and negative numbers, it can be expanded according to this more useful formula, for the first 4 terms:

((1+x)ⁿ = 1 + nx + \dfrac{n(n-1)}{2!} x^2 + \dfrac{n(n-1)(n-2)}{3!} x^3 + ... )

Partial fractions are sometimes used to generate expressions for binomial expansion.

[b]uValidity of binomial expansion[/u]/b

For ((p+qx)^n), when (n) is any rational number, the above binomial expansion formula is only bvalid/b when (\Big|\dfrac{qx}{p}\Big|<1).

((1+x)ⁿ = 1 + nx + \dfrac{n(n-1)}{2!} x^2 + \dfrac{n(n-1)(n-2)}{3!} x^3 + ... )

For ((p+qx)^n), when (n) is any rational number, the above binomial expansion formula is only valid when (\Big|\dfrac{qx}{p}\Big|<1).