I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$ and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$. (See https://en.wikipedia.org/wiki/Geometric_series#Sum)
From Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^i}$ it seems that there is no closed formula known for sums of the form $\sum_{k=0}^n r^{2^k}$, or more generally $\sum_{k=0}^n r^{s^k}$
I'm wondering about sums of the form $\sum_{k=0}^n r^{(k^2)}$, or more generally $\sum_{k=0}^n r^{(k^s)}$. Does anyone know of a formula for those sums?
A closed formula for me would mean the number of arithmetical operations it uses is not depending on $n$ and it uses only elementary functions (whatever that means to you).
Does anyone have an idea how one could prove there is no such formula?
