Almost convergent sequence

A bounded real sequence $(x_n)$ is said to be almost convergent to $L$ if each Banach limit assigns the same value $L$ to the sequence $(x_n)$.

Lorentz proved that $(x_n)$ is almost convergent if and only if

$\lim {}_{p\to \mathrm{\infty }}\frac{x_n+\dots +x_{n+p-1}}{p}=L$

uniformly in $n$.

The above limit can be rewritten in detail as

$\mathrm{\forall }\epsilon >0:\mathrm{\exists }{p}_0:\mathrm{\forall }p>p_0:\mathrm{\forall }n:|\frac{x_n+\dots +x_{n+p-1}}{p}-L|<\epsilon .$

Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.

References

• G. Bennett and N.J. Kalton: "Consistency theorems for almost convergence." Trans. Amer. Math. Soc., 198:23–43, 1974.
• J. Boos: "Classical and modern methods in summability." Oxford University Press, New York, 2000.
• J. Connor and K.-G. Grosse-Erdmann: "Sequential definitions of continuity for real functions." Rocky Mt. J. Math., 33(1):93–121, 2003.
• G.G. Lorentz: "A contribution to the theory of divergent sequences." Acta Math., 80:167–190, 1948.
• Hardy, G. H. (1949), Divergent Series, Oxford: Clarendon Press, https://archive.org/details/divergentseries033523mbp .

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