Series expansion: $1/(1-x)^n$

I realize this is an old thread, but I wanted to expand on the above answers on how to derive the formula for anyone else that might come along. Starting with the geometric series and taking successive derivatives:

$$ \begin{align} \dfrac{1}{(1-x)} &= 1+x+x^2+x^3+x^4+x^5\dots+x^m+\dotsm\\ \dfrac{1}{(1-x)^2} &= 1+2x+3x^2+4x^3+5x^4\dots+mx^{m-1}+\dotsm\\ \dfrac{2\cdot 1}{(1-x)^3} &= 2+(3\cdot 2)x+(4\cdot 3)x^2+(5\cdot 4)x^3\dots+(m \cdot m-1)x^{m-2}+\dotsm\\ \vdots\\ \dfrac{(n-1)!}{(1-x)^n} &= \sum_{k=0}^\infty \dfrac{(k+n-1)!}{k!} x^k\\ \end{align} $$

which can be simplified by dividing:

$$ \dfrac{1}{(1-x)^n} = \sum_{k=0}^\infty \dfrac{(k+n-1)!}{(n-1)!k!} x^k = \sum_{k=0}^\infty \binom{k+n-1}{n-1}x^k\\ $$

Yes its the binomial expansion for any index.

$(1-x)^{-n} = (-x)^{0} + -n(-x)^{1}+ \dfrac{-n(-n-1)}{2!}(-x)^{2} + ...$

which simplifies to ..

$(1-x)^{-n} = 1 + nx+ \dfrac{n(n+1)}{2!}(x)^{2} + \dfrac{n(n+1)(n+2)}{3!}(x)^{3} ...$

ie,

$(1-x)^{-n} = 1 + nx+ {n+1\choose 2}(x)^{2} + {n+2\choose 3}(x)^{3} ...$

Binomial expansion for any index is generalization of binomial theorem for positive integral index:

$$(1+x)^n = {n\choose 0} + {n\choose 1}x + {n\choose 2}x^2 + ...$$

There's a simpler version of the above formula:

$$\frac{1}{(1-x)^n}=\sum_{k=0}^\infty \binom{k+n-1}{n-1}x^k$$

You can prove this by induction - differentiate and then divide by $n$.