Periodic sequence

In mathematics, a periodic sequence (sometimes called a cycle or orbit) is a sequence for which the same terms are repeated over and over:

a₁, a₂, ..., a_p,  a₁, a₂, ..., a_p,  a₁, a₂, ..., a_p, ...

The number p of repeated terms is called the period (period).^[1]

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]} 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.^{[citation needed ]} The smallest p for which a periodic sequence is p-periodic is called its least period^[1] or exact period.

Every constant function is 1-periodic.

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

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

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

More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).^[4]

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

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

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

is a periodic sequence. Here, ${\displaystyle f^{n}(x)}${\displaystyle f^{n}(x)} 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.

${\displaystyle \sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n},\qquad \prod _{n=1}^{kp+m}a_{n}={\biggl (}{\prod _{n=1}^{p}a_{n}}{\biggr )}^{k}\cdot \prod _{n=1}^{m}a_{n}}${\displaystyle \sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n},\qquad \prod _{n=1}^{kp+m}a_{n}={\biggl (}{\prod _{n=1}^{p}a_{n}}{\biggr )}^{k}\cdot \prod _{n=1}^{m}a_{n}},

where ${\displaystyle m<p}${\displaystyle m<p} and ${\displaystyle k}${\displaystyle k} are positive integers.

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:

${\displaystyle \sum _{k=0}^{0}\cos \left(2\pi {\frac {nk}{1}}\right)/1=1,1,1,1,1,1,1,1,1,\cdots }${\displaystyle \sum _{k=0}^{0}\cos \left(2\pi {\frac {nk}{1}}\right)/1=1,1,1,1,1,1,1,1,1,\cdots }

${\displaystyle \sum _{k=0}^{1}\cos \left(2\pi {\frac {nk}{2}}\right)/2=1,0,1,0,1,0,1,0,1,0,\cdots }${\displaystyle \sum _{k=0}^{1}\cos \left(2\pi {\frac {nk}{2}}\right)/2=1,0,1,0,1,0,1,0,1,0,\cdots }

${\displaystyle \sum _{k=0}^{2}\cos \left(2\pi {\frac {nk}{3}}\right)/3=1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,\cdots }${\displaystyle \sum _{k=0}^{2}\cos \left(2\pi {\frac {nk}{3}}\right)/3=1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,\cdots }

${\displaystyle \cdots }${\displaystyle \cdots }

${\displaystyle \sum _{k=0}^{N-1}\cos \left(2\pi {\frac {nk}{N}}\right)/N=1,0,0,0,\cdots ,1,\cdots \quad {\text{sequence with period }}N}${\displaystyle \sum _{k=0}^{N-1}\cos \left(2\pi {\frac {nk}{N}}\right)/N=1,0,0,0,\cdots ,1,\cdots \quad {\text{sequence with period }}N}

One standard approach for proving these identities is to apply De Moivre's formula to the corresponding root of unity. Such sequences are foundational in the study of number theory.

A sequence is eventually periodic or ultimately periodic^[1] if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as ${\displaystyle a_{k+r}=a_{k}}${\displaystyle a_{k+r}=a_{k}} for some r and sufficiently large k. 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

${\displaystyle \lim _{n\rightarrow \infty }x_{n}-a_{n}=0.}${\displaystyle \lim _{n\rightarrow \infty }x_{n}-a_{n}=0.}^[3]

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, ....

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