In the bionomial expansion (1+x)^p

[suku]

In the bionomial expansion (1+x)^p , p can be integer or fraction. The coefficients are written as p
C
j . For this to hold p>= j. (j= integer). What happens if p=1/2,-1/2 or any other fraction? How can one use the same combination coefficient formula?

tks for any hlp.

 

[CompuChip]

Is that true?

As far as I know, you cannot write
$$\sqrt{1 + x} = (1 + x)^{1/2}$$
as
$$a 1^{1/2} + b x^{1/2} = a + b \sqrt{x}$$
for any numbers a, b, for example.

You can write it as
$$\sqrt{1 + x} = a_0 + a_1 x + a_2 x^2 + \cdots$$
but not with a finite series and a_n then have little to do with binomial coefficients.

The fact that it works with integers, is simply because you can open the brackets in, say, (1 + x)ⁿ = (1 + x)(1 + x)...(1 + x) [n times], and the coefficient of 1^k x^n-k is simply the number of ways in which you can choose k of the brackets whose 1 you multiply with the other n - k brackets' x, which is by definition n choose k.

 

[CRGreathouse]

What happens if p=1/2,-1/2 or any other fraction?

You need infinitely many terms in that case. Just continue to generate terms the usual way.