A sequence indexed from zero with an integer period satisfying the displayed identity for all . Repeating a finite cycle is an example. The Cesàro mean article explains why its cumulative averages approach the one-cycle arithmetic mean.
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A periodic sequence is a sequence of numbers that repeats itself after a certain number of terms. More formally, a sequence \((a_n)\) is considered periodic with period \(p\) if there exists a positive integer \(p\) such that for all integers \(n\): \[ a_{n + p} = a_n \] for all \(n\). This means that after every \(p\) terms, the sequence returns to the same value.