Almost convergent sequence explained

(x _{n)} is said to be almost convergent to L if each Banach limit assignsthe same value L to the sequence (x _{n)} .

Lorentz proved that

(x _{n)} is almost convergent if and only if\lim\limits _{p\toinfty} (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| (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 methodwhich 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.
• .

Specific

1. Hardy, p. 52