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]
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Let's say I've got a sequence. It starts at 1, then let's say it goes to negative 1/2. Then it goes to positive 1/3. Then it goes to negative 1/4. Then it goes to positive 1/5. And it just keeps going on and on and on like this. And we could graph it. Let me draw our vertical axis. So I'll graph this as our y-axis. And I'm going to graph y is equal to a sub n. And let's make this our horizontal axis where I'm going to plot our n's. So this right over here is our n's. And this is, let's say this right over here is positive 1. This right over here is negative 1. This would be negative 1/2. This would be positive 1/2. And I'm not drawing the vertical and horizontal axes at the same scale, just so that we can kind of visualize this properly. But let's say this is 1, 2, 3, 4, 5, and I could keep going on and on and on. So we see here that when n is equal to 1, a sub n is equal to 1. So this is right over there. So when n is equal to 1, a sub n is equal to 1. So this is y is equal to a sub n. Now, when n is equal to 2, we have a sub n is equal to negative 1/2. When n is equal to 3, a sub n is equal to 1/3, which is right about there. When n is equal to 4, a sub n is equal to negative 1/4, which is right about there. And then when n is equal to 5, a sub n is equal to positive 1/5, which is maybe right over there. And we keep going on and on and on. So you see the points, they kind of jump around, but they seem to be getting closer and closer and closer to 0. Which would make us ask a very natural question-- what happens to a sub n as n approaches infinity? Or another way of saying that is, what is the limit-- let me do this in a new color-- of a sub n as n approaches infinity? Well, let's think about if we can define a sub n explicitly. So we can define this sequence as a sub n where n starts at 1 and goes to infinity with a sub n equaling-- what does it equal? Well, if we ignore sign for a second, it looks like it's just 1 over n. But then we seem like we oscillate in signs. We start with a positive, then a negative, positive, negative. So we could multiply this times negative 1 to the-- let's see. If we multiply it times negative 1 to the n, then this one would be negative and this would be positive. But we don't want it that way. We want the first term to be positive. So we say negative 1 to the n plus 1 power. And you can verify this works. When n is equal to 1, you have 1 times negative 1 squared, which is just 1, and it'll work for all the rest. So we could write this as equaling negative 1 to the n plus 1 power over n. And so asking what the limit of a sub n as n approaches infinity is equivalent to asking what is the limit of negative 1 to the n plus 1 power over n as n approaches infinity is going to be equal to? Remember, a sub n, this is just a function of n. It's a function where we're limited right over here to positive integers as our domain. But this is still just a limit as something approaches infinity. And I haven't rigorously defined it yet, but you can conceptualize what's going on here. As n approaches infinity, the numerator is going to oscillate between positive and negative 1, but this denominator is just going to get bigger and bigger and bigger and bigger. So we're going to get really, really, really, really small numbers. And so this thing right over here is going to approach 0. Now, I have not proved this to you yet. I'm just claiming that this is true. But if this is true-- so let me write this down. If true, if the limit of a sub n as n approaches infinity is 0, then we can say that a sub n converges to 0. That's another way of saying this right over here. If it didn't, if the limit as n approaches infinity didn't go to some value right here-- and I haven't rigorously defined how we define a limit-- but if this was not true, if we could not set some limit-- it doesn't have to be equal to 0. As long as it-- if this was not equal to some number, then we would say that a sub n diverges.
Definition
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
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 : X → X 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
Partial Sums
- Where k and m<p are natural numbers.
Partial Products
- 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:
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 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
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.