Asymptotic sequence
= Asymptotic sequence
Functions $(\phi_n)$ form an asymptotic sequence at $x_0$ when $\phi_{n+1}=o(\phi_n)$. The notation
$$
f\sim\sum_{n\geq0}a_n\phi_n
$$
means that for every $N\geq0$,
$$
f-\sum_{n=0}^Na_n\phi_n=o(\phi_N).
$$
Successive division of the remainder by $\phi_N$ uniquely recovers every coefficient.