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]

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.^[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 $1,2,1,2,1,2\dots$ 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).^[4]

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 $f : X \to X$ is a point $x$ 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 $n$ -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

\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

\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:

\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^[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 $a_{k+r}=a_{k}$ 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.

^[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

^ ^a ^b ^c ^d "Ultimately periodic sequence", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
^ Bosma, Wieb. "Complexity of Periodic Sequences" (PDF). www.math.ru.nl. Retrieved 13 August 2021.
^ ^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.
^ Hosch, William L. (1 June 2018). "Rational number". Encyclopedia Britannica. Retrieved 13 August 2021.

[:0-1] "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.

[:2-3] 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.

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