Antilimits are used to apply the methods of sequence transformations 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.

Antilimits are used to apply the methods of sequence transformations 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.

Many more applications of antilimits in sequence transformations can be found in Avram Sidi's Practical Extrapolation Methods.

Antilimits are used to apply the methods of sequence transformations to divergent series. A classic example (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.

Antilimits are used to apply the methods of sequence transformations 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.