Let $$\sum_{k=0}^n b^{\alpha^{k}}$$ be a sum where $b \in \mathbb N_+$ and $\alpha \in \mathbb R, \alpha > 1$. What is its name and how can I calculate its closed form?
\begin{align} \sum_{k=0}^n b^{\alpha^k} & = b^{\alpha^0} + b^{\alpha^1}+ b^{\alpha^2} + b^{\alpha^3}+\dots+ b^{\alpha^n} \\ & = b + b^{\alpha}+ b^{\alpha\cdot \alpha}+ b^{\alpha\cdot \alpha\cdot \alpha}+ \dots+ b^{\alpha\cdot \alpha\cdot \ldots \cdot\alpha} \end{align}