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, ..., ap,  a1, a2, ..., ap,  a1, a2, ..., ap, ...

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

Definition

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

an+p = an

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.

Examples

Every constant function is 1-periodic.

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

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

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

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:

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

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 : XX 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. 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.

Identities

Partial Sums

${\displaystyle \sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n))$ Where k and m<p are natural numbers.

Partial Products

${\displaystyle \prod _{n=1}^{kp+m}a_{n}=({\prod _{n=1}^{p}a_{n)))^{k}*\prod _{n=1}^{m}a_{n))$ Where k and m<p are natural numbers.

Periodic 0, 1 sequences

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:

${\displaystyle \sum _{k=1}^{1}\cos \left(-\pi {\frac {n(k-1)}{1))\right)/1=1,1,1,1,1,1,1,1,1,\cdots }$
${\displaystyle \sum _{k=1}^{2}\cos \left(2\pi {\frac {n(k-1)}{2))\right)/2=0,1,0,1,0,1,0,1,0,\cdots }$
${\displaystyle \sum _{k=1}^{3}\cos \left(2\pi {\frac {n(k-1)}{3))\right)/3=0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,\cdots }$
${\displaystyle \cdots }$
${\displaystyle \sum _{k=1}^{N}\cos \left(2\pi {\frac {n(k-1)}{N))\right)/N=0,0,0,\cdots ,1,\cdots \quad {\text{sequence with period ))N}$

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.

Generalizations

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 ${\displaystyle a_{k+r}=a_{k))$ 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 x1x2x3, ... is asymptotically periodic if there exists a periodic sequence a1a2a3, ... for which

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

References

1. ^ a b c d "Ultimately periodic sequence", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. ^ Bosma, Wieb. "Complexity of Periodic Sequences" (PDF). www.math.ru.nl. Retrieved 13 August 2021.
3. ^ a b Janglajew, Klara; Schmeidel, Ewa (2012-11-14). "Periodicity of solutions of nonhomogeneous linear difference equations". Advances in Difference Equations. 2012 (1): 195. doi:10.1186/1687-1847-2012-195. ISSN 1687-1847. S2CID 122892501.
4. ^ Hosch, William L. (1 June 2018). "Rational number". Encyclopedia Britannica. Retrieved 13 August 2021.