In mathematics, a **sequence** is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called *elements*, or *terms*). The number of elements (possibly infinite) is called the *length* of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.

- Examples and notation
- Examples
- Indexing
- Defining a sequence by recursion
- Formal definition and basic properties
- Definition
- Finite and infinite
- Increasing and decreasing
- Bounded
- Subsequences
- Other types of sequences
- Limits and convergence
- Formal definition of convergence
- Applications and important results
- Cauchy sequences
- Infinite limits
- Series
- Use in other fields of mathematics
- Topology
- Analysis
- Linear algebra
- Abstract algebra
- Set theory
- Computing
- Streams
- See also
- Notes
- References
- External links

For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be * finite *, as in these examples, or * infinite *, such as the sequence of all even positive integers (2, 4, 6, ...).

The position of an element in a sequence is its *rank* or *index*; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of , and , where the subscript *n* refers to the *n*th element of the sequence; for example, the *n*th element of the Fibonacci sequence * is generally denoted as **.*

In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

A sequence can be thought of as a list of elements with a particular order.^{ [1] }^{ [2] } Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers.

There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting a sequence are discussed after the examples.

The prime numbers are the natural numbers bigger than 1 that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics, particularly in number theory where many results related to them exist.

The Fibonacci numbers comprise the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).^{ [1] }

Other examples of sequences include those made up of rational numbers, real numbers and complex numbers. The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as the limit of a sequence of rational numbers (e.g. via its decimal expansion). As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π, that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that is easily discernible by inspection.

The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences.^{ [3] }

Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of π. One such notation is to write down a general formula for computing the *n*th term as a function of *n*, enclose it in parentheses, and include a subscript indicating the set of values that *n* can take. For example, in this notation the sequence of even numbers could be written as . The sequence of squares could be written as . The variable *n* is called an index, and the set of values that it can take is called the index set.

It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like , which denotes a sequence whose *n*th element is given by the variable . For example:

One can consider multiple sequences at the same time by using different variables; e.g. could be a different sequence than . One can even consider a sequence of sequences: denotes a sequence whose *m*th term is the sequence .

An alternative to writing the domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation denotes the ten-term sequence of squares . The limits and are allowed, but they do not represent valid values for the index, only the supremum or infimum of such values, respectively. For example, the sequence is the same as the sequence , and does not contain an additional term "at infinity". The sequence is a **bi-infinite sequence**, and can also be written as .

In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes for an arbitrary sequence. Often, the index *k* is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in

In some cases, the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways.

Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be the natural numbers. In the second and third bullets, there is a well-defined sequence , but it is not the same as the sequence denoted by the expression.

Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion. This is in contrast to the definition of sequences of elements as functions of their positions.

To define a sequence by recursion, one needs a rule, called *recurrence relation* to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation.

The Fibonacci sequence is a simple classical example, defined by the recurrence relation

with initial terms and . From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34.

A complicated example of a sequence defined by a recurrence relation is Recamán's sequence,^{ [4] } defined by the recurrence relation

with initial term

A *linear recurrence with constant coefficients* is a recurrence relation of the form

where are constants. There is a general method for expressing the general term of such a sequence as a function of n; see Linear recurrence. In the case of the Fibonacci sequence, one has and the resulting function of n is given by Binet's formula.

A holonomic sequence is a sequence defined by a recurrence relation of the form

where are polynomials in n. For most holonomic sequences, there is no explicit formula for expressing explicitly as a function of n. Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions.

Not all sequences can be specified by a recurrence relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...).

There are many different notions of sequences in mathematics, some of which (*e.g.*, exact sequence) are not covered by the definitions and notations introduced below.

In this article, a sequence is formally defined as a function whose domain is an interval of integers. This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring the domain of a sequence to be the set of natural numbers. This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition, the codomain of the sequence is fixed by context, for example by requiring it to be the set **R** of real numbers,^{ [5] } the set **C** of complex numbers,^{ [6] } or a topological space.^{ [7] }

Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, *a _{n}* rather than

Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets. A **net** is a function from a (possibly uncountable) directed set to a topological space. The notational conventions for sequences normally apply to nets as well.

The **length** of a sequence is defined as the number of terms in the sequence.

A sequence of a finite length *n* is also called an *n*-tuple. Finite sequences include the **empty sequence** ( ) that has no elements.

Normally, the term *infinite sequence* refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence is called a **singly infinite sequence** or a **one-sided infinite sequence** when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a **bi-infinite sequence**, **two-way infinite sequence**, or **doubly infinite sequence**. A function from the set **Z** of *all* integers into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ), is bi-infinite. This sequence could be denoted .

A sequence is said to be * monotonically increasing * if each term is greater than or equal to the one before it. For example, the sequence is monotonically increasing if and only if *a*_{n+1}*a*_{n} for all *n* ∈ **N**. If each consecutive term is strictly greater than (>) the previous term then the sequence is called **strictly monotonically increasing**. A sequence is **monotonically decreasing**, if each consecutive term is less than or equal to the previous one, and **strictly monotonically decreasing**, if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a **monotone** sequence. This is a special case of the more general notion of a monotonic function.

The terms **nondecreasing** and **nonincreasing** are often used in place of *increasing* and *decreasing* in order to avoid any possible confusion with *strictly increasing* and *strictly decreasing*, respectively.

If the sequence of real numbers (*a _{n}*) is such that all the terms are less than some real number

A ** subsequence ** of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of the positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved.

Formally, a subsequence of the sequence is any sequence of the form , where is a strictly increasing sequence of positive integers.

Some other types of sequences that are easy to define include:

- An
**integer sequence**is a sequence whose terms are integers. - A
**polynomial sequence**is a sequence whose terms are polynomials. - A positive integer sequence is sometimes called
**multiplicative**, if*a*_{nm}=*a*_{n}*a*_{m}for all pairs*n*,*m*such that*n*and*m*are coprime.^{ [8] }In other instances, sequences are often called*multiplicative*, if*a*_{n}=*na*_{1}for all*n*. Moreover, a*multiplicative*Fibonacci sequence^{ [9] }satisfies the recursion relation*a*_{n}=*a*_{n−1}*a*_{n−2}. - A binary sequence is a sequence whose terms have one of two discrete values, e.g. base 2 values (0,1,1,0, ...), a series of coin tosses (Heads/Tails) H,T,H,H,T, ..., the answers to a set of True or False questions (T, F, T, T, ...), and so on.

An important property of a sequence is *convergence*. If a sequence converges, it converges to a particular value known as the *limit*. If a sequence converges to some limit, then it is **convergent**. A sequence that does not converge is **divergent**.

Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value (called the limit of the sequence), and they become and remain *arbitrarily* close to , meaning that given a real number greater than zero, all but a finite number of the elements of the sequence have a distance from less than .

For example, the sequence shown to the right converges to the value 0. On the other hand, the sequences (which begins 1, 8, 27, …) and (which begins −1, 1, −1, 1, …) are both divergent.

If a sequence converges, then the value it converges to is unique. This value is called the **limit** of the sequence. The limit of a convergent sequence is normally denoted . If is a divergent sequence, then the expression is meaningless.

A sequence of real numbers **converges to** a real number if, for all , there exists a natural number such that for all we have^{ [5] }

If is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that denotes the complex modulus, i.e. . If is a sequence of points in a metric space, then the formula can be used to define convergence, if the expression is replaced by the expression , which denotes the distance between and .

If and are convergent sequences, then the following limits exist, and can be computed as follows:^{ [5] }^{ [10] }

- for all real numbers
- , provided that
- for all and

Moreover:

- If for all greater than some , then .
^{ [lower-alpha 1] } - (Squeeze Theorem)

If is a sequence such that for all and ,

then is convergent, and . - If a sequence is bounded and monotonic then it is convergent.
- A sequence is convergent if and only if all of its subsequences are convergent.

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is *Cauchy characterization of convergence for sequences*:

- A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.

In contrast, there are Cauchy sequences of rational numbers that are not convergent in the rationals, e.g. the sequence defined by *x*_{1} = 1 and *x*_{n+1} = *x*_{n} + 2/*x*_{n}/2 is Cauchy, but has no rational limit, cf. here. More generally, any sequence of rational numbers that converges to an irrational number is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers.

Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called complete metric spaces and are particularly nice for analysis.

In calculus, it is common to define notation for sequences which do not converge in the sense discussed above, but which instead become and remain arbitrarily large, or become and remain arbitrarily negative. If becomes arbitrarily large as , we write

In this case we say that the sequence **diverges**, or that it **converges to infinity**. An example of such a sequence is *a*_{n} = *n*.

If becomes arbitrarily negative (i.e. negative and large in magnitude) as , we write

and say that the sequence **diverges** or **converges to negative infinity**.

A **series** is, informally speaking, the sum of the terms of a sequence. That is, it is an expression of the form or , where is a sequence of real or complex numbers. The **partial sums** of a series are the expressions resulting from replacing the infinity symbol with a finite number, i.e. the *N*th partial sum of the series is the number

The partial sums themselves form a sequence , which is called the **sequence of partial sums** of the series . If the sequence of partial sums converges, then we say that the series is **convergent**, and the limit is called the **value** of the series. The same notation is used to denote a series and its value, i.e. we write .

Sequences play an important role in topology, especially in the study of metric spaces. For instance:

- A metric space is compact exactly when it is sequentially compact.
- A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences.
- A metric space is a connected space if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.
- A topological space is separable exactly when there is a dense sequence of points.

Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems to spaces without metrics.

The topological product of a sequence of topological spaces is the cartesian product of those spaces, equipped with a natural topology called the product topology.

More formally, given a sequence of spaces , the product space

is defined as the set of all sequences such that for each *i*, is an element of . The ** canonical projections ** are the maps *p _{i}* :

In analysis, when talking about sequences, one will generally consider sequences of the form

which is to say, infinite sequences of elements indexed by natural numbers.

It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by *x _{n}* = 1/log(

The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field *K*, where *K* is either the field of real numbers or the field of complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in *K*, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

The most important sequences spaces in analysis are the ℓ^{p} spaces, consisting of the *p*-power summable sequences, with the *p*-norm. These are special cases of L^{p} spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted *c* and *c*_{0}, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called an FK-space.

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of *F*-valued sequences (where *F* is a field) is a function space (in fact, a product space) of *F*-valued functions over the set of natural numbers.

Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or rings.

If *A* is a set, the free monoid over *A* (denoted *A*^{*}, also called Kleene star of *A*) is a monoid containing all the finite sequences (or strings) of zero or more elements of *A*, with the binary operation of concatenation. The free semigroup *A*^{+} is the subsemigroup of *A*^{*} containing all elements except the empty sequence.

In the context of group theory, a sequence

of groups and group homomorphisms is called **exact**, if the image (or range) of each homomorphism is equal to the kernel of the next:

The sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms.

In homological algebra and algebraic topology, a **spectral sequence** is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by JeanLeray ( 1946 ), they have become an important research tool, particularly in homotopy theory.

An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and *X* is a set, an α-indexed sequence of elements of *X* is a function from α to *X*. In this terminology an ω-indexed sequence is an ordinary sequence.

In computer science, finite sequences are called lists. Potentially infinite sequences are called streams. Finite sequences of characters or digits are called strings.

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as *sequences* or * streams *, as opposed to finite * strings *. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0, 1}). The set *C* = {0, 1}^{∞} of all infinite binary sequences is sometimes called the Cantor space.

An infinite binary sequence can represent a formal language (a set of strings) by setting the *n* th bit of the sequence to 1 if and only if the *n* th string (in shortlex order) is in the language. This representation is useful in the diagonalization method for proofs.^{ [11] }

- Operations

- Examples

- Discrete-time signal
- Farey sequence
- Fibonacci sequence
- Look-and-say sequence
- Thue–Morse sequence
- List of integer sequences

- Types

- ±1-sequence
- Arithmetic progression
- Automatic sequence
- Cauchy sequence
- Constant-recursive sequence
- Geometric progression
- Harmonic progression
- Holonomic sequence
- Regular sequence
- Pseudorandom binary sequence
- Random sequence

- Related concepts

- List (computing)
- Net (topology) (a generalization of sequences)
- Ordinal-indexed sequence
- Recursion (computer science)
- Set (mathematics)
- Tuple
- Permutation

- ↑ If the inequalities are replaced by strict inequalities then this is false: There are sequences such that for all , but .

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, a **Cauchy sequence**, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

In mathematics, a **series** is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

In mathematics, **real analysis** is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, the **limit inferior** and **limit superior** of a sequence can be thought of as limiting bounds on the sequence. They can be thought of in a similar fashion for a function. For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called **infimum limit**, **limit infimum**, **liminf**, **inferior limit**, **lower limit**, or **inner limit**; limit superior is also known as **supremum limit**, **limit supremum**, **limsup**, **superior limit**, **upper limit**, or **outer limit**.

In mathematics, a **power series** is an infinite series of the form

In mathematics, and specially in algebra, a **formal series** is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series.

In mathematics, an infinite series of numbers is said to **converge absolutely** if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to **converge absolutely** if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if

In the mathematical field of real analysis, the **monotone convergence theorem** is any of a number of related theorems proving the convergence of monotonic sequences that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In mathematics, the **limit of a sequence** is the value that the terms of a sequence "tend to", and is often denoted using the symbol. If such a limit exists, the sequence is called **convergent**. A sequence that does not converge is said to be **divergent**. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.

In mathematics, **pointwise convergence** is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.

In mathematics, the **limit** of a sequence of sets is a set whose elements are determined by the sequence in either of two equivalent ways: **(1)** by upper and lower bounds on the sequence that converge monotonically to the same set and **(2)** by convergence of a sequence of indicator functions which are themselves real-valued. As is the case with sequences of other objects, convergence is not necessary or even usual.

The **Arzelà–Ascoli theorem** is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

In mathematics, a **divergent series** is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field *K* of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in *K*, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted

In number theory, **natural density** is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval [1, *n*] as *n * grows large.

In mathematics, **convergence tests** are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .

In mathematical analysis, the **spaces of test functions and distributions** are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the *canonical LF-topoogy*, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called * the space of distributions on * and is denoted by where the "" subscript indicates that the continuous dual space of denote by is endowed with the strong dual topology.

- 1 2 "Sequences".
*www.mathsisfun.com*. Retrieved 2020-08-17. - ↑ Weisstein, Eric W. "Sequence".
*mathworld.wolfram.com*. Retrieved 2020-08-17. - ↑ Index to OEIS, On-Line Encyclopedia of Integer Sequences, 2020-12-03
- ↑ Sloane, N. J. A. (ed.). "SequenceA005132(Recamán's sequence)".
*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 26 January 2018. - 1 2 3 Gaughan, Edward (2009). "1.1 Sequences and Convergence".
*Introduction to Analysis*. AMS (2009). ISBN 978-0-8218-4787-9. - ↑ Edward B. Saff & Arthur David Snider (2003). "Chapter 2.1".
*Fundamentals of Complex Analysis*. ISBN 978-01-390-7874-3. - ↑ James R. Munkres (2000). "Chapters 1&2".
*Topology*. ISBN 978-01-318-1629-9. - ↑ Lando, Sergei K. (2003-10-21). "7.4 Multiplicative sequences".
*Lectures on generating functions*. AMS. ISBN 978-0-8218-3481-7. - ↑ Falcon, Sergio (2003). "Fibonacci's multiplicative sequence".
*International Journal of Mathematical Education in Science and Technology*.**34**(2): 310–315. doi:10.1080/0020739031000158362. S2CID 121280842. - ↑ Dawikins, Paul. "Series and Sequences".
*Paul's Online Math Notes/Calc II (notes)*. Retrieved 18 December 2012. - ↑ Oflazer, Kemal. "FORMAL LANGUAGES, AUTOMATA AND COMPUTATION: DECIDABILITY" (PDF).
*cmu.edu*. Carnegie-Mellon University. Retrieved 24 April 2015.

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