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?
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 $|x|<a^{-1}.$