Periodic sequence

Summary

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 edit

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 edit

Every constant function is 1-periodic.

The sequence   is periodic with least period 2.

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

 

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:

 

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

 

is a periodic sequence. Here,   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 edit

Partial Sums edit

  Where k and m<p are natural numbers.

Partial Products edit

  Where k and m<p are natural numbers.

Periodic 0, 1 sequences edit

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:

 
 
 
 
 

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 edit

A sequence is eventually periodic if it can be made periodic by dropping some finite number of terms from the beginning. 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 ultimately periodic if it satisfies the condition   for some r and sufficiently large k.[1]

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

 [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 edit

  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.