Antilimits are used to apply the methods of <A HREF="http://en.wikipedia.org/wiki/Sequence_transformation">sequence transformations</A> to divergent series. There is no unique definition, but typically if the sum $\sum_{n}a_n(x)$ converges to some function $f(x)$ for $|x|<\rho$, and this function can be continued analytically for $|x|>\rho$, then $f(x)$ is called the *limit* of the divergent series for $|x|<\rho$ and the *antilimit* for $|x|>\rho$.

A classic application (from Christopher Small's *Expansions and Asymptotics for Statistics*) is the proof of the identity

$$\frac{1}{1-x}+\frac{1}{1+x^2}-\frac{x}{1-x^2}=\frac{2}{1-x^4}$$

by sequence transformations:

$$\frac{1}{1-x}+\frac{1}{1+x^2}-\frac{x}{1-x^2}=(1+x+x^2+\cdots)+$$
$$(1-x^2+x^4-\cdots)-(x+x^3+x^5+\cdots)$$
$$=2+2x^4+2x^8+\cdots=\frac{2}{1-x^4}.$$

Although the sums of the series only have a limit for $|x|<1$, the proof of the identity remains valid for $|x|>1$ if the sums are interpreted as antilimits.