Geometric-mean refinement of an asymptotic sequence
= Geometric-mean refinement of an asymptotic sequence
For $\chi_0=\phi_0$ and $\chi_n=\sqrt{\phi_{n-1}\phi_n}$, the sequence $(\chi_n)$ is asymptotic, but $\chi_n/\phi_n\to\infty$ for every $n\geq1$.