Asymptotic sequence

A sequence of functions $\{\phi_n(x)\}$ such that

$$\phi_{n+1}(x)=o(\phi_n(x)),\quad x\to x_0,\quad x\in M,$$

where $x_0$ is a limit point of the set $M$ (finite or infinite). If the nature of $M$ is clear from the context, then one simply writes $x\to x_0$. If $\{\phi_n(x)\}$ is an asymptotic sequence and $\psi(x)$ is a function defined on $M$, then $\{\psi(x)\phi_n(x)\}$ will also be an asymptotic sequence.

Examples of asymptotic sequences:

1) $\{(x-x_0)^n\},x\to x_0$;

2) $\{x^{-n}\},x\to\infty$;

3) $\{e^xx^{-n}\},x\to\infty$;

4) $\{z^{-n}\},z\to\infty,z\in D$, where $D$ is an unbounded domain in the complex plane. Asymptotic sequences such as 1), 2) and 4) are called asymptotic power sequences.

References

[a1]	N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
[a2]	A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)