= Asymptotic sequence
An <asymptotic sequence> orders the scales against which successive errors are compared. On a specified domain approaching $x_0$, suppose each $\phi_n$ is nonzero sufficiently close to the <limit> and
$$
\frac{\phi_{n+1}(x)}{\phi_n(x)}\longrightarrow0.
$$
Equivalently, $\phi_{n+1}=o(\phi_n)$ in <little o notation>. The notation
$$
f\sim\sum_{n\geq0}a_n\phi_n
$$
means that for every $N\geq0$,
$$
f-\sum_{n=0}^Na_n\phi_n=o(\phi_N).
$$
Each $N$ stays fixed while the parameter tends to its <limit>.<NIST Digital Library of Mathematical Functions/2.1 Definitions and Elementary Properties>[NIST Digital Library of Mathematical Functions, §2.1(v), generalized asymptotic scales and expansions]{ref}
As $z\to+\infty$, $1,z^{-1},z^{-2},\ldots$ is an <asymptotic sequence>, because each adjacent ratio is $1/z$. Fractional scales $1,z^{-1/2},z^{-1},\ldots$ work too, with ratio $z^{-1/2}$. By contrast, $1,1+z^{-1},z^{-1},\ldots$ fails at its first pair: the second scale is not smaller than the first. Ordering by written position alone does not create a hierarchy of sizes.
Successive division of the remainder by $\phi_N$ uniquely recovers every <coefficient>:
$$
a_N=\lim_{x\to x_0}
\frac{f(x)-\sum_{n=0}^{N-1}a_n\phi_n(x)}{\phi_N(x)}.
$$
If two proposed expansions first differ at order $N$, subtracting them and dividing by $\phi_N$ would give a nonzero constant tending to zero, a contradiction. This is uniqueness of the <coefficients> in a fixed scale, not uniqueness of the represented <function>.
For the factorial <integral> example, the first three limiting responses are $\lim F(z)=1$, $\lim z(F(z)-1)=-1$, and $\lim z^2(F(z)-1+z^{-1})=2$. They extract exactly the coefficients $1,-1,2!$. Choosing an <asymptotic sequence> therefore turns “the next small correction” into a precise and reusable operation.
Back to article page
