Solving summations of form $\sum_{i=0}^{\infty} i^kx^i$ without Differentiation and Integration

Question

The actual sum I'm looking to solve is $$\sum x^i(i^3-(i-1)^3)=(1-x)\sum x^ii^3$$ But I'd rather do a general answer first. I know the method of differentiation and integration but I'd like to do it without that. I remember seeing a method which reduced the power of $i$ on Brilliant but I can't find it anymore.

Answer 1

If $F_k(x) = \displaystyle\sum_{i = 0}^{\infty} i^k x^i$, then (assuming $0^0 = 1$) $F_0(x) = 1 / (1 - x)$ and, for $k > 0$, $$F_{k}(x) = \sum_{i = 0}^{\infty}(i + 1)^k x^{i + 1} = x \sum_{r = 0}^{k} \binom{k}{r}F_r(x),$$ which gives you $(1 - x)F_k(x)$ expressed through $F_r(x)$ with $r < k$.

This is just an example - there are other recurrences that you can get like this way.