I need to show that $\phi_n(z)=\ln(1+z^n)$ as $z \rightarrow 0$ is an asymptotic sequence, i.e. to show that $\lim_{z\rightarrow 0}\frac{\phi_{n+1}(z)}{\phi_n(z)}=0.$
Is it sufficient for me to say that as $z\rightarrow 0$, $\frac{\phi_{n+1}}{\phi_n}=\frac{\ln(1+z^{n+1})}{\ln(1+z^n)} \rightarrow 0?$ Because $z^{n+1}$ and $z^n \rightarrow 0$ as $z\rightarrow 0$, and $\ln(1)=0?$