Binomial theorem for any index (need not be a natural number)

If n is a rational number and x is a real number such that |x|<1, then

(1+x) ⁿ = 1 = nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3!+…+n(n-1)(n-2)…(n-r+1)x^r/r!+...+ ∞

Remarks

The condition |x|<1 is unnecessary when n is a whole number.

When n is not a whole number, then the condition |x|<1 is necessary.

The terms are infinite when n is not an whole number. When is it an whole number the series become finite as one of the terms will become zero in the coefficient at some point in time.

When n is a positive integer, there will be n+1 terms

To expand (x+a)ⁿ proceed as follows:



(x+a)ⁿ = {a(1 + x/a)} ⁿ

= aⁿ(1 + x/a) ⁿ; (substitute x’ = x/a and proceed)