 Asymptotic expansion

An asymptotic expansion describes a function near a specified limit by successively smaller scales. Each fixed finite truncation has a remainder smaller than its last retained scale. The central question is how the approximation improves as the parameter approaches the limit; convergence of the entire infinite series at a fixed parameter is a separate question. For the inverse-power scale $1,z^{-1},z^{-2},\dots$,means that, for every fixed nonnegative integer $m$,The little o notation makes the accuracy requirement precise. With expansions to all orders, this is also equivalent to an $O(z^{-m-1})$ remainder after the $m$th term; the implied bound may depend on $m$.^[ref]

For a quick convergent example, the finite geometric series identity givesFor $z>1$, both the asymptotic expansion as $z\to +\infty$ and the ordinary convergent geometric series describe this function. The exact remainder displays the required extra inverse power.

The more surprising example under a divergent factorial series can approximate a convergent integral has a convergent integral but a divergent series at every fixed positive $z$:A few terms can estimate $F(10)$ well, even though adding sufficiently many terms eventually destroys the approximation. Optimal truncation explains where to stop.

This makes asymptotic expansions useful when an exact expression is a complicated integral or differential equation solution, but a parameter is very large or very small. The leading term reveals the scale; corrections reveal which effects matter next. The asymptotic sequence must be specified: integer powers, fractional powers, logarithms, and exponential functions can encode different kinds of behavior. A good finite approximation includes a remainder estimate or an independently checked range of use.