Editing 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 [[term (logic)|terms]] are repeated over and over:

:''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''p''</sub>,&nbsp;&nbsp;''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''p''</sub>,&nbsp;&nbsp;''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''p''</sub>, ...

The number ''p'' of repeated terms is called the '''period''' ([[frequency|period]]).<ref name=":0">{{eom|title=Ultimately periodic sequence}} </ref>

==Definition==
A '''(purely) periodic''' sequence (with '''period ''p'''''), or a '''''p-''periodic sequence''', is a sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... satisfying

:''a''<sub>''n''+''p''</sub> = ''a''<sub>''n''</sub>
for all values of ''n''.<ref name=":0" /><ref>{{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}}</ref><ref name=":2">{{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}}</ref> If a sequence is regarded as a [[function (mathematics)|function]] whose domain is the set of [[natural number]]s, 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'''<ref name=":0" /> or '''exact period'''.

==Examples==
Every constant function is 1-periodic.

The sequence <math>1,2,1,2,1,2\dots</math> is periodic with least period 2.

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

:<math>\frac{1}{7} = 0.142857\,142857\,142857\,\ldots</math>

More generally, the sequence of digits in the decimal expansion of any [[rational number]] is eventually periodic (see below).<ref>{{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}}</ref>

The sequence of powers of &minus;1 is periodic with period two:

:<math>-1,1,-1,1,-1,1,\ldots</math>

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 (group theory)|order]] in a [[group (mathematics)|group]].

A [[periodic point]] for a function {{math|''f'' : ''X'' → ''X''}} is a point {{mvar|x}} whose [[orbit (dynamics)|orbit]]

:<math>x,\, f(x),\, f(f(x)),\, f^3(x),\, f^4(x),\, \ldots</math>

is a periodic sequence.  Here, <math>f^n(x)</math> means the {{nowrap|{{mvar|n}}-fold}} [[Function composition|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 ==
:<math>\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}</math>, 

where <math>m < p</math> and <math>k</math> 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:

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

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

:<math>\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</math>

:<math>\cdots</math>

:<math>\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 </math>

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'''<ref name=":0" /> if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as  <math>a_{k+r} = a_k</math> 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&nbsp;&nbsp;8 5 7 1 4 2&nbsp;&nbsp;8 5 7 1 4 2&nbsp;&nbsp;8 5 7 1 4 2&nbsp;&nbsp;...

A sequence is '''asymptotically periodic''' if its terms approach those of a periodic sequence.  That is, the sequence ''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;''x''<sub>3</sub>,&nbsp;... is asymptotically periodic if there exists a periodic sequence ''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;''a''<sub>3</sub>,&nbsp;... for which

:<math>\lim_{n\rightarrow\infty} x_n - a_n = 0.</math><ref name=":2" />

For example, the sequence

:1 / 3,&nbsp;&nbsp;2 / 3,&nbsp;&nbsp;1 / 4,&nbsp;&nbsp;3 / 4,&nbsp;&nbsp;1 / 5,&nbsp;&nbsp;4 / 5,&nbsp;&nbsp;...

is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....

== References ==
{{Reflist}}{{Series (mathematics)}}

{{DEFAULTSORT:Periodic Sequence}}
[[Category:Sequences and series]]