Almost convergent sequence

A bounded real sequence ( x n ) {\displaystyle (x_{n})} is said to be almost convergent to L {\displaystyle L} if each Banach limit assigns the same value L {\displaystyle L} to the sequence ( x n ) {\displaystyle (x_{n})} . Lorentz proved that ( x n ) {\displaystyle (x_{n})} is almost convergent if and only if lim p → ∞ x n + … + x n + p − 1 p = L {\displaystyle \lim \limits _{p\to \infty }{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}=L} uniformly in n {\displaystyle n} .

Source: Wikipedia — Almost convergent sequence (CC BY-SA 4.0)

Almost convergent sequence

A bounded real sequence ( x n ) {\displaystyle (x_{n})} is said to be almost convergent to L {\displaystyle L} if each Banach limit assigns the same value L {\displaystyle L} to the sequence ( x n ) {\displaystyle (x_{n})} . Lorentz proved that ( x n ) {\displaystyle (x_{n})} is almost convergent if and only if lim p → ∞ x n + … + x n + p − 1 p = L {\displaystyle \lim \limits _{p\to \infty }{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}=L} uniformly in n {\displaystyle n} .

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Source: Wikipedia "Almost convergent sequence" · CC BY-SA 4.0

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