Definition:Asymptotic Series

Definition

An asymptotic series is a divergent series of the form:

$a_0 + \dfrac {a_1} x + \dfrac {a_1} {x_2} + \cdots + \dfrac {a_n} {x_n} + \cdots$

where $a_0, a_1, a_2, \ldots$ are constants.

This is an asymptotic representation of a function $\map f x$ if:

$\forall n \in \N: \ds \lim_{\size x \mathop \to \infty} x^n \paren {\map f x - \map {s_n} x} = 0$

where $s_n$ is the sum of the first $n$ terms.

Also see

• Results about asymptotic series can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): asymptotic series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): asymptotic series