TITLE: What series does the function $\frac{1}{(1-ax)^r}$ generate?
QUESTION [1 upvotes]: I want to know what series the function
$$1/(1-ax)^r, \quad a,r\in \mathbb{N}, $$
generates.
I thought about doing this: Let's name $y=ax$. Now we have
$$\frac{1}{(1-y)^r}, \quad r\in \mathbb{N},$$
and we know that
$$\frac{1}{(1-y)^r}= \sum_{n=0}^{\infty}{n+r-1\choose r-1}y^n.$$
Now let's put back $y=ax$. So
$$\frac{1}{(1-ax)^r} = \sum_{n=0}^{\infty}{n+r-1\choose r-1}a^nx^n.$$
Does this make sense?

REPLY [1 votes]: The series is $$S=(1-ax)^{-r}=\sum_{k=0}^{\infty}(-1)^k {-r \choose k}  (ax)^k=\sum_{k=0}^{\infty}  {r+k-1 \choose r - 1} (ax)^k.$$
Which is valid for $|x|<a^{-1}.$
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