Binomial Expansion/Theorem?

[apple2357]

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??

 

[Dr.Peterson]

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??

Did you try searching deeper? I found Newton's generalized binomial theorem, which is what you are asking about, and that led to this link in Wikipedia.

I hope you recognize that you can't literally use the same formula for the general case, but have to adapt it, as shown on these pages.

 

[apple2357]

Did you try searching deeper? I found Newton's generalized binomial theorem, which is what you are asking about, and that led to this link in Wikipedia.

I hope you recognize that you can't literally use the same formula for the general case, but have to adapt it, as shown on these pages.

Thank you for this. I will have a read. Yes I recognise that substituting anything other than a positive integer for n or r or zero into the nCr is problematic. But aometimes I have seen that expanded for example nC2 is (n-1) (n-2)/2! And then substituted in. Is that allowed?

 

[Dr.Peterson]

Thank you for this. I will have a read. Yes I recognise that substituting anything other than a positive integer for n or r or zero into the nCr is problematic. But aometimes I have seen that expanded for example nC2 is (n-1) (n-2)/2! And then substituted in. Is that allowed?

Yes, that's what you'll see on both pages I linked to.