# Limit of radii of convergence of growth series

Consider the Coxeter group $$G_n$$ generated by a finite set $$\{s_1, ..., s_n\}$$ with respect to the relations $$s_1^2=...=s_n^2=1$$ and $$s_is_j=s_js_i$$ for $$|i-j| \geq 2$$ and denote the word length with respect to the generating set by $$l$$. The growth series of $$G_n$$ is defined as $$\sum_{g\in G_n} z^{l(g)}$$. Denote its radius of convergence by $$\rho_n$$. Is it then true that $$\rho_n \rightarrow 0$$ as $$n\rightarrow \infty$$? Intuitively this seems to be clear but I encounter problems trying to prove this.