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 \limits _{{p\to \infty }}{\frac {x_{{n}}+\ldots +x_{{n+p-1}}}p}=L

uniformly in $n$ .

The above limit can be rewritten in detail as

\forall \varepsilon >0:\exists p_{0}:\forall p>p_{0}:\forall n:\left|{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}-L\right|<\varepsilon .

Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.^[1]

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 .

Specific

↑ Hardy,p.52

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[1] Hardy,p.52