Asymptotic sequence

Functions $({\varphi }_n)$ form an asymptotic sequence at $x_0$ when ${\varphi }_{n+1}=o({\varphi }_n)$. The notationmeans that for every $N\ge 0$,Successive division of the remainder by ${\varphi }_N$ uniquely recovers every coefficient.

For a positive asymptotic sequence $({\varphi }_n)$, define ${\psi }_0={\varphi }_0$ andThen ${\psi }_n\sim {\varphi }_n$ term by term and $({\psi }_n)$ is again an asymptotic sequence.

For ${\chi }_0={\varphi }_0$ and ${\chi }_n=\sqrt{{\varphi }_{n-1}{\varphi }_n}$, the sequence $({\chi }_n)$ is asymptotic, but ${\chi }_n/{\varphi }_n\to \infty$ for every $n\ge 1$.

Even if ${\varphi }_n\sim {\psi }_n$ for every $n$, an expansion in the scale $({\varphi }_n)$ need not induce one in $({\psi }_n)$. A perturbation of ${\psi }_0$ that is smaller than ${\varphi }_0$ but larger than ${\varphi }_1$ supplies a counterexample.