For $a,b\in\mathbb{R}^+$ we define $$ \alpha_i = \frac{2\pi}{2^ib}\quad \mathrm{ for }\quad i\in \mathbb N_0\qquad h_0=\frac{1}{a^2\alpha_0^3} $$ and $$ h_i=a\bigg (\alpha_i(h_i+\sum_{j=0}^{i-1}h_j) \bigg)^{3/2}\quad \mathrm{ for }\quad i\in \mathbb N $$ I'm only interested in values of the $h_i$, which are $\in \mathbb R^+$, since theses $h_i$ describe some heights in a physical problem.
My question now is if $$ \sum_{i=0}^\infty{h_i} $$ converges. If $(h_i)_{i\in\mathbb N}$ converges or its suppremum is not infinity, then I would already have a proof, but I can't seem to get either of these properties. Maybe though my approach is wrong alltogether.
Since the equation for $h_i$ is obtained through trying to describe a physical process, it might not be 100% well defined, but if I obtain solutions for the $h_i$ numerically then they definitely converge to $0$. I would like to have some concret mathematical proof though.
Maybe someone has an idea or tipp, thank you very much in advance!