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## Asymptotic expansion - Wikipedia

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## Asymptotic expansion

Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function.

Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics.

The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e. Such methods are often referred to as hyperasymptotic approximations. See asymptotic analysis and big O notation for the notation used in this article. First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.

L may be taken to be infinity. If f is a continuous function on the domain of the asymptotic scale, then f has an asymptotic expansion of order N with respect to the scale as a formal series. If one or the other holds for all N , then we write [ citation needed ]. Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. Thus, for example, one may start with the ordinary series.