Finding the coefficient of $x^n $ in a rational expression

Question

For a homework question, I need to find the coefficient of $x^n$ in $\frac{-x^7}{4(1-x)^2}$.

I've never done anything like this, and my teacher said he will take it up in tomorrow class, but I'm really curious as to how I can do this.

I know that I need to use the negative binomial expression, but I'm not sure how to deal with the $x^7$ term in the numerator.

Any help to get me started would be great. Thanks

Answer 1

Coefficient of $x^{n}$ in $\frac 1 4 (-x^{7}) (1+x+x^{2}+\cdots)(1+x+x^{2}+\cdots)$ is $-\frac 1 4(1+1+\cdots+1)$ where there are $(n-7+1)$ one's. So the answer is $-\frac 1 4 (n-7+1)$for $n \geq 7$. It is $0$ for $n <7$.

Answer 2

Since $(1-x)^{-2} =\sum_{n=0}^{\infty} (n+1)x^n $, $x^m(1-x)^{-2} =\sum_{n=0}^{\infty} (n+1)x^{n+m} =\sum_{n=m}^{\infty} (n-m+1)x^{n} $, so the coefficient of $x^n$ is $n-m+1$.

For $m=7$ this is $n-7+1 = n-6$.

Since your divides this by $-4$, the answer is $-(n-6)/4$.