How can I multiply these sums? How do I multiply this? Is it even possible given that one sum is infinite?
$$
x^{n}\left[\,\sum_{r = 0}^{n}\left(-1\right)^{r}x^{mr}\,\right]
\left[\,\sum_{k = 0}^{\infty}{n + k - 1 \choose k} x^{k}\,\right]
$$

The first factor with the finite alternating geometric sum over $r$ is $$\frac{1-(-x^m)^{n+1}}{x^m+1}$$. The second infinite sum over $k$ is $\frac{1}{(1-x)^n}$.
