"Solving Coefficient of x^n in Binomial Expansion

[thereddevils]

Homework Statement

Find, in the simplest form, the coefficient of x^n in the binomial expansion of (1-x)^(-6).

Homework Equations

The Attempt at a Solution

i am not sure how to go about with this.

 

[Willian93]

are u sure you have to find a cofficient contain x^n?

because u should have a specific value for n, so that you could find the cofficient infront of it, or you should at least have which term you are looking for.

 

[thereddevils]

are u sure you have to find a cofficient contain x^n?

because u should have a specific value for n, so that you could find the cofficient infront of it, or you should at least have which term you are looking for.

Yes, that's the question. Maybe it's asking for the coefficient of for any term in the expression.

 

[HallsofIvy]

Since the exponent, -6, is not a positive integer, you will need to use the generalized binomial series:
$$(a+ b)^m= \sum_{k=0}^\infty \frac{m(m-1)\cdot\cdot\cdot(m-k+1)}{k!}a^kb^{m-k}$$

Here, of course, a= 1, b= -x, and m= -6 so this is

$$(1- x)^{-6}= \sum_{k=0}^\infty \frac{-6(-7)\cdot\cdot\cdot(-5-k)}{k!}(-1)^kx^{-6-k}$$

You, apparently, are asked for the coefficient when -6-k= n or when k= -6-n.

 

[thereddevils]

Since the exponent, -6, is not a positive integer, you will need to use the generalized binomial series:
$$(a+ b)^m= \sum_{k=0}^\infty \frac{m(m-1)\cdot\cdot\cdot(m-k+1)}{k!}a^kb^{m-k}$$

Here, of course, a= 1, b= -x, and m= -6 so this is

$$(1- x)^{-6}= \sum_{k=0}^\infty \frac{-6(-7)\cdot\cdot\cdot(-5-k)}{k!}(-1)^kx^{-6-k}$$

You, apparently, are asked for the coefficient when -6-k= n or when k= -6-n.

thanks

Is this answer the most simplified?

$$\frac{-6(-7)...(-11-n)}{(-6-n)!}$$

the general formula for binomial series for (a+b)^n is different when n is a positive integer and when n is a fractional or negative value?

$$(a+ b)^m= \sum_{k=0}^\infty \frac{m(m-1)\cdot\cdot\cdot(m-k+1)}{k!}a^kb^{m-k}$$

Does it matter if the powers(k and n-k) for a and b is swapped since a+b is commutative?

This is the continuation of the question:

Hence, find the coefficient of x^6 and x^7 in (1+2x+3x^2+4x^3+5x^4+6x^5+7x^6)^3

 

[thereddevils]

any further hints on this question?