Periodic sequence

In mathematics, a periodic sequence (sometimes called a cycle) 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).

Definition

A periodic sequence is a sequence a1, a2, a3, ... satisfying

an+p = an

for all values of n. If we regard a sequence as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.

Examples

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

1 / 7 = 0 . 1 4 2 8 5 7  1 4 2 8 5 7  1 4 2 8 5 7  ...

More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).

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

−1, +1, −1, +1, −1, +1, ...

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 ƒ: X → X is a point p whose orbit

p,\, f(p),\, f(f(p)),\, f^3(p),\, f^4(p),\, \ldots

is a periodic sequence. 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.

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=1}^{k=1} \cos (-2\pi\frac{n(k-1)}{1})/1 = 1,1,1,1,1,1,1,1,1...
\sum_{k=1}^{k=2} \cos (-2\pi\frac{n(k-1)}{2})/2 = 0,1,0,1,0,1,0,1,0...
\sum_{k=1}^{k=3} \cos (-2\pi\frac{n(k-1)}{3})/3 = 0,0,1,0,0,1,0,0,1,0,0,1,0,0,1...
...
\sum_{k=1}^{k=N} \cos (-2\pi\frac{n(k-1)}{N})/N = 0,0,0...,1 \text{ sequence with period } N


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 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, ....