I know how to derive the formula $\sum_{k=0}^\infty q^k = \frac{1}{1-q}$ for $|q| < 1$. I also recall having found the respective formulae for $\sum_{k=0}^\infty k q^k$ and $\sum_{k=0}^\infty k^2 q^k$ rewriting e.g. $\sum_{k=0}^\infty k q^k = q \frac{\mathrm d}{\mathrm dq} \sum_{k=0}^\infty q^k.$
Now I recall having proven the first formula (for $\sum_{k=0}^\infty q^k$) before having discussed functions (in particular derivatives and integrals), that is to say right when I started to learn about infinite series. However, I've only come across the second and third formula near the end of Calculus 1, thus only deriving them using derivatives.
My question is: Is it possible to derive the formulae with less knowledge, particularly not using any derivatives or integrals? If yes, does the same apply for higher powers ($\geq 3$) of $k$?
Thanks for your answers in advance.