Can someone help clarify some confusion?
I can accept ( and develop, prove even ) the Binomial expansion for positive n:
(x + y)^n = nC0 0 x^n y^0 + nC1 1 x^n - 1 y^1 + nC2 2 x^n-2 y^2 +.....+ y^n
The series will always terminate.
How do we know we can use this formula with negative/ rational n?
We often see
(1+x)^-1 generated using the above formula to give = 1+x+x^2+ .... etc
This using n=-1 and generates an infinite polynomial.
I have done some searching and can't see a proof of why the first formula can be used for negative or rational n??
I can accept ( and develop, prove even ) the Binomial expansion for positive n:
(x + y)^n = nC0 0 x^n y^0 + nC1 1 x^n - 1 y^1 + nC2 2 x^n-2 y^2 +.....+ y^n
The series will always terminate.
How do we know we can use this formula with negative/ rational n?
We often see
(1+x)^-1 generated using the above formula to give = 1+x+x^2+ .... etc
This using n=-1 and generates an infinite polynomial.
I have done some searching and can't see a proof of why the first formula can be used for negative or rational n??