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- a1, a2, ..., ap, a1, a2, ..., ap, a1, a2, ..., ap, ...
The number p of repeated terms is called the period (period).
A periodic sequence is a sequence a1, a2, a3, ... satisfying
- an+p = an
The sequence of digits in the decimal expansion of 1/7 is periodic with period 6:
More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).
The sequence of powers of −1 is periodic with period two:
is a periodic sequence. Here, means the n-fold composition of f applied to x. Periodic points are important in the theory of dynamical systems. Every function from a finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point.
Periodic 0, 1 sequencesEdit
Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions:
A sequence is eventually periodic if it can be made periodic by dropping some finite number of terms from the beginning. For example, the sequence of digits in the decimal expansion of 1/56 is eventually periodic:
- 1 / 56 = 0 . 0 1 7 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 ...
A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence x1, x2, x3, ... is asymptotically periodic if there exists a periodic sequence a1, a2, a3, ... for which
For example, the sequence
- 1 / 3, 2 / 3, 1 / 4, 3 / 4, 1 / 5, 4 / 5, ...
is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....