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 ${\displaystyle 1,2,1,2,1,2\dots }$ 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 : X → X 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 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 ${\displaystyle a_{k+r}=a_{k}}$ 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 x₁, x₂, x₃, ... is asymptotically periodic if there exists a periodic sequence a₁, a₂, a₃, ... 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

• v
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Sequences and series

Integer sequences

Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced (list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Properties of sequences

• Cauchy sequence
• Monotonic function
• Periodic sequence

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series

Hypergeometric series

• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series

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