Periodic sequence

A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a₁, a₂, a₃, ... satisfying

a_n+p = a_n

for all values of n.[1][2][3][4][5] If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function. The smallest p for which a periodic sequence is p-periodic is called its least period[1][6] or exact period.[6]

Every constant function is 1-periodic.[4]

The sequence ${\displaystyle 1,2,1,2,1,2\dots }$ is periodic with least period 2.[2]

The sequence of digits in the decimal expansion of 1/7 is periodic with period 6:

${\displaystyle {\frac {1}{7}}=0.142857\,142857\,142857\,\ldots }$

More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).[7]

The sequence of powers of −1 is periodic with period two:

${\displaystyle -1,1,-1,1,-1,1,\ldots }$

More generally, the sequence of powers of any root of unity is periodic. The same holds true for the powers of any element of finite order in a group.

A periodic point for a function f : X → X is a point x whose orbit

${\displaystyle x,\,f(x),\,f(f(x)),\,f^{3}(x),\,f^{4}(x),\,\ldots }$

is a periodic sequence. Here, ${\displaystyle f^{n}(x)}$ means the n-fold composition of f applied to x.[6] 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.

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:

${\displaystyle \sum _{k=1}^{1}\cos(-\pi {\frac {n(k-1)}{1}})/1=1,1,1,1,1,1,1,1,1...}$

${\displaystyle \sum _{k=1}^{2}\cos(2\pi {\frac {n(k-1)}{2}})/2=0,1,0,1,0,1,0,1,0...}$

${\displaystyle \sum _{k=1}^{3}\cos(2\pi {\frac {n(k-1)}{3}})/3=0,0,1,0,0,1,0,0,1,0,0,1,0,0,1...}$

${\displaystyle ...}$

${\displaystyle \sum _{k=1}^{N}\cos(2\pi {\frac {n(k-1)}{N}})/N=0,0,0...,1{\text{ sequence with period }}N}$

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 ultimately periodic if it satisfies the condition ${\displaystyle a_{k+r}=a_{k}}$ for some r and sufficiently large k.[1]

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

${\displaystyle \lim _{n\rightarrow \infty }x_{n}-a_{n}=0.}$[4][8][9]

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, ....