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Published in:
2010 | OriginalPaper | Chapter
Asymptotic Expansions
Author : Kenneth Lange
Published in: Numerical Analysis for Statisticians
Publisher: Springer New York
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Asymptotic analysis is a branch of mathematics dealing with the order of magnitude and limiting behavior of functions, particularly at boundary points of their domains of definition [1, 2, 4, 5, 7]. Consider, for instance, the function
$$f(x) = \frac{x^2 + 1}{x + 1}.$$
It is obvious that
f
(
x
) resembles the function
x
as
$$x \rightarrow \infty$$
. However, one can be more precise. The expansion
$$\begin{array}{rcl}f(x) &&= \frac{x^2 + 1}{x(1 + \frac{1}{x})}\\ &&= \left(x + \frac{1}{x}\right)\sum\limits_{k = 0}^{\infty}\left(\frac{-1}{x}\right)^{k}\\ &&= x - 1 - 2\sum\limits_{k = 1}^{\infty}\left(\frac{-1}{x}\right)^{k}\end{array}$$
indicates that
f
(
x
) more closely resembles
x
− 1 for large
x
. Furthermore,
f
(
x
) −
x
+ 1 behaves like 2/
x
for large
x
. We can refine the precision of the approximation by taking more terms in the infinite series. How far we continue in this and other problems is usually dictated by the application at hand.