Asymptotic Expansion
An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for f. The idea is that successive terms provide a more and more accurate description of the order of growth of f. An example is Stirling's approximation.
In symbols, it means we have
but also
and
for each fixed k, while some limit is taken, usually with the requirement that gk+1 = o(gk), which means the (gk) form an asymptotic scale. The requirement that the successive sums improve the approximation may then be expressed as
In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. However, this optimal partial sum will usually have more terms as the argument approaches the limit value.
Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method, saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). The famous Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.
Read more about this topic: Asymptotic Analysis
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