periodic sequence

{{short description|Sequence for which the same terms are repeated over and over}}

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).{{eom|title=Ultimately periodic sequence}}

Definition

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.{{Cite web|last=Bosma|first=Wieb|title=Complexity of Periodic Sequences|url=https://www.math.ru.nl/~bosma/pubs/periodic.pdf|access-date=13 August 2021|website=www.math.ru.nl}}{{Cite journal|last1=Janglajew|first1=Klara|last2=Schmeidel|first2=Ewa|date=2012-11-14|title=Periodicity of solutions of nonhomogeneous linear difference equations|journal=Advances in Difference Equations|volume=2012|issue=1|pages=195|doi=10.1186/1687-1847-2012-195|s2cid=122892501|issn=1687-1847|doi-access=free}} 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|date=August 2021}} The smallest p for which a periodic sequence is p-periodic is called its least period or exact period.

Examples

Every constant function is 1-periodic.

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

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

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

More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).{{Cite web|last=Hosch|first=William L.|date=1 June 2018|title=Rational number|url=https://www.britannica.com/science/rational-number|access-date=13 August 2021|website=Encyclopedia Britannica|language=en}}

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

: $-1,1,-1,1,-1,1,\ldots$

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 {{math|f : X → X}} is a point {{mvar|x}} whose orbit

: $x,\, f(x),\, f(f(x)),\, f^3(x),\, f^4(x),\, \ldots$

is a periodic sequence. Here, $f^n(x)$ means the {{nowrap|{{mvar|n}}-fold}} composition of {{mvar|f}} applied to {{mvar|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.

Partial sums and products

: $\sum_{n=1}^{kp+m} a_{n} = k*\sum_{n=1}^{p} a_{n} + \sum_{n=1}^{m} a_{n}, \qquad \prod_{n=1}^{kp+m} a_{n} = \biggl({\prod_{n=1}^{p} a_{n}}\biggr)^k \cdot \prod_{n=1}^{m} a_{n}$ ,

where $m < p$ and $k$ are positive integers.

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:

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

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

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

: $\cdots$

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

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.

Generalizations

A sequence is eventually periodic or ultimately periodic if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as $a_{k+r} = a_k$ 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

: $\lim_{n\rightarrow\infty} x_n - a_n = 0.$

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

Category:Sequences and series