Series expansion: $1/(1-x)^n$ What is the expansion for $(1-x)^{-n}$?
Could find only the expansion upto the power of $-3$. Is there some general formula?
 A: 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:
$$(1+x)^n = {n\choose 0} + {n\choose 1}x + {n\choose 2}x^2 + ...$$
A: 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 $n$.
A: 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:
$$
\dfrac{1}{(1-x)^n} = \sum_{k=0}^\infty \dfrac{(k+n-1)!}{(n-1)!k!} x^k = \sum_{k=0}^\infty \binom{k+n-1}{n-1}x^k\\
$$
A: As $\;\biggl(\dfrac1{1-x}\biggr)^{(n-1)}=\dfrac{(n-1)!}{(1-x)^n},\;$ you can derive term by term the power series expansion of $\;\dfrac1{1-x}=1+x+x^2+\dots+x^m+\dotsm$
You obtain 
\begin{align*}
\frac{(n-1)!}{(1-x)^n}&=\sum_{m\ge n-1}m(m-1)\dotsm(m-n+2)x^{m-n+1}\\ &=\sum_{m\ge n-1}\frac{m!}{(m-n+1)!}x^{m-n+1}
=\sum_{m\ge 0}\frac{(m+n-1)!}{m!}x^m,\\
\text{whence}\qquad \frac1{(1-x)^n}&=\sum_{m\ge 0}\binom{m+n-1}{m}x^m.
\end{align*}
A: I wrote this answer up to reduce my own feelings of uncertainty about the expansion.

