Sequence (mathematics)
A sequence in mathematics is an ordered list of mathematical objects, typically numbers, which can be either finite or infinite and allows for the repetition of terms. Unlike sets, where the order does not matter, the arrangement of terms in a sequence is crucial, meaning that sequences (1, 2) and (2, 1) are considered different. Sequences can be represented with indexed notation, mapping their position to their values. They can be defined explicitly using formulas, such as arithmetic sequences (a_n = cn + b) and geometric sequences (a_n = d • g_n), or recursively, as seen in the Fibonacci sequence where each term is the sum of the two preceding ones.
Understanding sequences is significant in both theoretical and applied mathematics, as they serve as the foundation for series, which are the summation of sequence terms. Sequences can converge, meaning they approach a specific value as they progress, or diverge, where no limit is reached. Applications of sequences range from simple counting and ordering tasks to complex modeling in various fields, including population growth and historical event analysis. Overall, sequences play a vital role in mathematics, facilitating a deeper understanding of numerical relationships and functions.
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Sequence (mathematics)
A sequence is an ordered list of mathematical objects, such as numbers, that may be infinite or finite and allows for repetition. A an important basic concept in mathematics, a sequence can take many forms and have various characteristics. They form the basis for series.
Finite sequences, or strings, have a definite number of terms with them, such as (1, −2, 7, π) and (0, 
, 0). Infinite sequences, or streams, contain an infinite number of terms, such as the sequence of natural numbers: (1, 2, 3, 4, 5…). Although both sequences and sets are collections of elements, they differ in two main ways. First, the order in which terms appear is significant in a sequence but not in a set. For example, while (1, 2) and (2, 1) represent different sequences, {1, 2} and {2, 1} are simply two ways of describing the set containing 1 and 2. Secondly, terms may be counted multiple times within a sequence, but only once within a set. The sequences (0, 0, 1, 2) and (0, 1, 2) are distinct, but the sets {0, 0, 1, 2} and {0, 1, 2} are equivalent.
Overview
Sequences are frequently indexed by listing each term in the form of an, where the subscript n is the position of the term within the sequence. In other words, a sequence is a function that maps n to an. As a result, the first term can be represented by a1 (or a0) the second by a2 (or a1), and so on. To avoid having to list out every term, an can be defined in terms of an explicit formula. For example, the sequence of natural numbers can be represented by an = n.
Certain sequences show up so often in pure or applied mathematics that they are given names. For instance, arithmetic sequences are sequences of the form an = cn + b, where c and b are constants. Geometric sequences have the form an = d •gn, where d and g are constants. Yet another type of sequence is a recursive sequence, in which terms are defined based on previous terms of the sequence. A notable historic example of a recursive sequence is the Fibonacci sequence, named after Leonardo Fibonacci of Pisa. In this sequence, each term is defined as the sum of the two previous terms, which can be expressed as follows:
The Fibonacci sequence is one example of an infinite sequence. When working with infinite sequences, it is often useful to find the limit of the sequence, the value an approaches as n becomes arbitrarily large. If an gets increasingly close or equal to a particular value k as n becomes larger, the sequence is said to "converge to k." If no such limit exists, the sequence is deemed divergent. One example of a convergent sequence is defined by an = 1/n, because as n approaches infinity, an approaches 0. Conversely, the Fibonacci Sequence and the sequence defined by an = n are divergent, since an approaches infinity as n does.
Applications of Sequences
Sequences can be applied to situations both in everyday life and in more advanced mathematical topics. Because any list of ordered numbers is a sequence, everything from prime numbers to population growth can be represented as a sequence. There is even a growing field called sequence analysis that uses sequences to model historical events.
One of the most important applications of sequences in higher mathematics are the summations of their terms, which are called series. Like sequences, series can be either finite or infinite, and infinite series can be either divergent or convergent. Convergent series are extremely useful in a variety of areas, but most importantly they can be used to calculate integrals that are impossible to find using other methods.
Bibliography
Blanchard, Philippe, Felix Bühlmann, and Jacques-Antoine Gauthier, eds. Advances in Sequence Analysis: Theory, Method, Applications. New York: Springer, 2014. Print.
Boyer, Carl B. "The Fibonacci Sequence." A History of Mathematics. 2nd ed. New York: Wiley, 1991. 255–56. Print.
Epp, Susana S. Discrete Mathematics and Applications. Boston: Cengage, 2011. Print.
Grabiner, Judith V. The Origins of Cauchy’s Rigorous Calculus. Mineola: Dover, 2011. Print.