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The '''surface area''' of a solid object is the total area of the object's faces and curved surfaces. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of [[arc length]] of one-dimensional curves, or of the surface area for [[polyhedra]] (i.e., objects with flat polygonal [[Face (geometry)|faces]]), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a [[sphere]], are assigned surface area using their representation as [[parametric surface]]s. This definition of surface area is based on methods of [[infinitesimal]] [[calculus]] and involves [[partial derivative]]s and [[double integration]].
{{Other uses}}


A general definition of surface area was sought by [[Henri Lebesgue]] and [[Hermann Minkowski]] at the turn of the twentieth century. Their work led to the development of [[geometric measure theory]], which studies various notions of surface area for irregular objects of any dimension. An important example is the [[Minkowski content]] of a surface.
In [[mathematics]], a '''sequence''' is an ordered list of objects (or events). Like a [[Set (mathematics)|set]], it contains [[Element (mathematics)|members]] (also called ''elements'', or ''terms''). The number of ordered elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a [[Discrete mathematics|discrete]] [[function (mathematics)|function]].


==Definition==
For example, (M, A, R, Y) is a sequence of letters that differs from (A, R, M, Y), as the ordering matters, and (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''[[finite set|finite]]'', as in this example, or ''[[Infinite set|infinite]]'', such as the sequence of all [[even and odd numbers|even]] [[positive and negative numbers|positive]] [[integer]]s (2, 4, 6,...).  Finite sequences are sometimes known as ''strings'' or ''words'' and infinite sequences as ''streams''.  The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
[[Image:Cauchy sequence illustration2.svg|right|thumb|350px|An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor [[Cauchy sequence|Cauchy]]. It is, however, bounded.]]


While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical ''definition'' of area requires a great deal of care.  
== Examples and notation ==
This should provide a function
There are various and quite different notions of sequences in mathematics, some of which (''e.g.'', [[exact sequence]]) are not covered by the notations introduced below.


: <math> S \mapsto A(S) </math>
In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names for convenient referencing.  For example a sequence might be written as (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, … ), or (''b''<sub>0</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>, … ), or (''c''<sub>0</sub>, ''c''<sub>2</sub>, ''c''<sub>4</sub>, … ), depending on what is useful in the application.


which assigns a positive [[real number]] to a certain class of [[surface]]s that satisfies several natural requirements. The most fundamental property of the surface area is its '''additivity''': ''the area of the whole is the sum of the areas of the parts''. More rigorously, if a surface ''S'' is a union of finitely many pieces ''S''<sub>1</sub>, …, ''S''<sub>''r''</sub> which do not overlap except at their boundaries, then
===Finite and infinite===
: <math> A(S) = A(S_1) + \cdots + A(S_r). </math>
A more formal definition of a '''finite sequence''' with terms in a set ''S'' is a [[function (mathematics)|function]] from  {1, 2, ..., ''n''} to ''S'' for some ''n'' > 0. An '''infinite sequence''' in ''S'' is a function from {1, 2, ... } to ''S''. For example, the sequence of prime numbers (2,3,5,7,11, … ) is the function 1→'''2''', 2→'''3''', 3→'''5''', 4→'''7''', 5→'''11''', … .


Surface areas of flat polygonal shapes must agree with their geometrically defined [[area]]. Since surface area is a geometric notion, areas of [[congruence (geometry)|congruent]] surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the [[Euclidean group|group of Euclidean motions]]. These properties uniquely characterize surface area for a wide class of geometric surfaces called ''piecewise smooth''. Such surfaces consist of finitely many pieces that can be represented in the [[parametric surface|parametric form]]
A sequence of a finite length ''n'' is also called an [[n-tuple|''n''-tuple]]. Finite sequences include the '''empty sequence''' ( ) that has no elements.


: <math> S_D: \vec{r}=\vec{r}(u,v), \quad (u,v)\in D </math>
A function from ''all'' integers into a set is sometimes called a '''bi-infinite sequence''' or '''two-way infinite sequence'''.  An example is the bi-infinite sequence of all even integers ( … , -4, -2, 0, 2, 4, 6, 8… ).


with a [[continuously differentiable]] function <math>\vec{r}.</math> The area of an individual piece is defined by the formula
===Multiplicative===
Let ''A'' = (a sequence defined by a function ''f'':{1, 2, 3, ...} → {1, 2, 3, ...}, such that ''a''<sub> ''i''</sub> = ''f''(''i'').


: <math> A(S_D) = \iint_D\left |\vec{r}_u\times\vec{r}_v\right | \, du \, dv. </math>
The sequence is '''multiplicative''' if ''f''(''xy'') = ''f''(''x'')''f''(''y'') for all ''x'',''y''  such that ''x'' and ''y'' are [[coprime]].<ref>{{cite book|title=Lectures on generating functions|last=Lando|first=Sergei K.|publisher=AMS|ISBN=0-8218-3481-9|chapter=7.4 Multiplicative sequences}}</ref>


Thus the area of ''S''<sub>''D''</sub> is obtained by integrating the length of the normal vector <math>\vec{r}_u\times\vec{r}_v</math> to the surface over the appropriate region ''D'' in the parametric ''uv'' plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs ''z'' = ''f''(''x'',''y'') and [[surface of revolution|surfaces of revolution]].
==Types and properties==
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.


One of the subtleties of surface area, as compared to [[arc length]] of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by [[Hermann Schwarz]] that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area (Known as [[Schwarz's paradox]].)
If the terms of the sequence are a subset of an [[partially ordered set|ordered set]], then a ''monotonically increasing'' sequence is one for which each term is greater than or equal to the term before it; if each term is [[strict]]ly greater than the one preceding it, the sequence is called ''strictly monotonically increasing''. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the [[monotonic function|monotonicity]] property is called monotonic or ''monotone''. This is a special case of the more general notion of [[monotonic function]].
<ref name=sch1>http://www.math.usma.edu/people/Rickey/hm/CalcNotes/schwarz-paradox.pdf</ref>
.<ref name=sch2>http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf</ref>


Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by [[Henri Lebesgue]] and [[Hermann Minkowski]]. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of [[fractal]]s. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in [[geometric measure theory]]. A specific example of such an extension is the [[Minkowski content]] of the surface.
The terms ''nondecreasing'' and ''nonincreasing'' are used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively.


== Common formulas ==
If the terms of a sequence are [[integer]]s, then the sequence is an  [[integer sequence]]. If the terms of a sequence are [[polynomial]]s, then the sequence is a [[polynomial sequence]].


{| class="wikitable"
If ''S'' is endowed with a [[topology]], then it becomes possible to consider ''convergence'' of an infinite sequence in ''S''. Such considerations involve the concept of the [[limit of a sequence]].
|+ Surface areas of common solids
|-
!Shape
!Equation
!Variables
|-
|[[Cube]]
|<math> 6s^2 \, </math>
|''s'' = side length
|-
|[[Rectangular prism]]
|<math> 2(\ell w + \ell h + wh) \, </math>
|''ℓ'' = length, ''w'' = width, ''h'' = height
|-
|[[Triangular prism]]
|<math> bh + l(a + b + c) </math>
|''b'' = base length of triangle, ''h'' = height of triangle, ''l'' = distance between triangles, ''a'', ''b'', ''c'' = sides of triangle
|-
|All [[Prism (geometry)|Prisms]]
|<math> 2B + Ph \, </math>
|''B'' = the area of one base, ''P'' = the perimeter of one base, ''h'' = height
|-
|[[Sphere]]
|<math> 4\pi r^2 = \pi d^2\, </math>
|''r'' = radius of sphere, ''d'' = diameter
|-
|[[Spherical lune]]
|<math> 2r^2\theta \, </math>
|''r'' = radius of sphere, ''θ'' = [[dihedral angle]]
|-
|[[Torus]]
|<math> (2\pi r)(2\pi R) = 4\pi^2 Rr</math>
|''r'' = minor radius, ''R'' = major radius
|-
|Closed [[Cylinder (geometry)|cylinder]]
|<math> 2\pi r^2 + 2\pi rh = 2\pi r(r+h) \, </math>
|''r'' = radius of the circular base, ''h'' = height of the cylinder
|-
|Lateral surface area of a [[cone (geometry)|cone]]
|<math> \pi r \left(\sqrt{r^2+h^2}\right) = \pi rs \, </math>
|<math> s = \sqrt{r^2+h^2} </math><br>
''s'' = slant height of the cone,<br>
''r'' = radius of the circular base,<br>
''h'' = height of the cone
|-
|Full surface area of a cone
|<math> \pi r \left(r + \sqrt{r^2+h^2}\right) = \pi r(r + s) \, </math>
| ''s'' = slant height of the cone,<br>
''r'' = radius of the circular base,<br>
''h'' = height of the cone
|-
|[[Pyramid (geometry)|Pyramid]]
|<math>B + \frac{PL}{2}</math>
|''B'' = area of base, ''P'' = perimeter of base, ''L'' = slant height
|-
|[[Square pyramid]]
|<math> b^2 + 2bs </math>
|''b'' = base length, ''s'' = slant height
|}


===Ratio of surface areas of a sphere and cylinder of the same Radius and Height===
If A is a set, the [[free monoid]] over A (denoted A<sup>*</sup>) is a [[monoid]] containing all the finite sequences (or strings) of zero or more elements drawn from A, with the binary operation of concatenation.  The [[free semigroup]] A<sup>+</sup> is the subsemigroup of A<sup>*</sup> containing all elements except the empty sequence.


[[Image:Inscribed cone sphere cylinder.svg|thumb|300px|A cone, sphere and cylinder of radius ''r'' and height ''h''.]]
==Analysis==
The below given formulas can be used to show that the surface area of a [[sphere]] and [[cylinder (geometry)|cylinder]] of the same radius and height are in the ratio '''2&nbsp;:&nbsp;3''', as follows.
In [[mathematical analysis|analysis]], when talking about sequences, one will generally consider sequences of the form
:<math>(x_1, x_2, x_3, \dots)\text{ or }(x_0, x_1, x_2, \dots)\,</math>
which is to say, infinite sequences of elements indexed by [[natural number]]s.


Let the radius be ''r'' and the height be ''h'' (which is 2''r'' for the sphere).
It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by  ''x<sub>n</sub>'' = 1/[[logarithm|log]](''n'') would be defined only for ''n'' ≥ 2.
When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices [[large enough]], that is, greater than some given ''N''.


<math>\begin{array}{rlll}
The most elementary type of sequences are numerical ones, that is, sequences of real or [[complex number]]s.
\text{Sphere surface area}  & = 4 \pi r^2      &                    & = (2 \pi r^2) \times 2 \\
This type can be generalized to sequences of elements of some [[vector space]]. In analysis, the vector spaces considered are often [[function space]]s. Even more generally, one can study sequences with elements in some [[topological space]].
\text{Cylinder surface area} & = 2 \pi r (h + r) & = 2 \pi r (2r + r) & = (2 \pi r^2) \times 3
\end{array}</math>


The discovery of this ratio is credited to [[Archimedes]].<ref>{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|title = Tomb of Archimedes: Sources|publisher = Courant Institute of Mathematical Sciences|accessdate = 2007-01-02}}</ref>
==Series==
{{main|Series (mathematics)}}
The sum of terms of a sequence is a [[series (mathematics)|series]]. More precisely, if (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ...) is a sequence, one may consider the sequence of [[partial sum]]s (''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ...), with


== In chemistry ==
:<math>S_n=x_1+x_2+\cdots + x_n=\sum\limits_{i=1}^{n}x_i.</math>
{{see also|Accessible surface area}}
Surface area is important in [[chemical kinetics]]. Increasing the surface area of a substance generally increases the [[reaction rate|rate]] of a [[chemical reaction]]. For example, [[iron]] in a fine powder will [[combustion|combust]], while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired.


== In biology ==
Formally, this pair of sequences comprises the ''series'' with the terms ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ..., which is denoted as
{{see also|Surface-area-to-volume ratio}}
[[Image:Mitochondrion 186.jpg|right|thumb|The inner membrane of the [[mitochondrion]] has a large surface area due to infoldings, allowing higher rates of [[cellular respiration]] (electron [[micrograph]]).]]
The surface area of an organism is important in several considerations, such as regulation of body temperature and [[digestion]]. Animals use their [[teeth]] to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains [[microvilli]], greatly increasing the area available for absorption. [[Elephant]]s have large [[ear]]s, allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss.


The [[surface area to volume ratio]] (SA:V) of a [[cell (biology)|cell]] imposes upper limits on size, as the volume increases much faster than does the surface area, thus limiting the rate at which substances diffuse from the interior across the [[cell membrane]] to interstitial spaces or to other cells. Indeed, representing a cell as an idealized [[sphere]] of radius ''r'', the volume and surface area are, respectively, ''V'' = 4/3 π ''r''<sup>3</sup>; ''SA'' = 4 π ''r''<sup>2</sup>.  The resulting surface area to volume ratio is therefore 3/''r''. Thus, if a cell has a radius of 1 μm, the SA:V ratio is 3; whereas if the radius of the cell is instead 10 μm, then the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Thus, the surface area falls off steeply with increasing volume.
:<math>\sum\limits_{i=1}^\infty x_i.</math>


== References ==
If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see [[series (mathematics)|series]].


<references />
==Infinite sequences in theoretical computer science==
Infinite sequences of [[numerical digit|digits]] (or [[character (computing)|characters]]) drawn from a [[finite set|finite]] [[alphabet (computer science)|alphabet]] are of particular interest in [[theoretical computer science]]. They are often referred to simply as ''sequences'' or ''streams'', as opposed to finite ''[[String (computer science)#Formal theory|strings]]''. Infinite binary sequences, for instance, are infinite sequences of [[bit]]s (characters drawn from the alphabet {0,1}). The set ''C'' = {0, 1}<sup>∞</sup> of all infinite, binary sequences is sometimes called the [[Cantor space]].


* {{eom|title=Area|id=A/a013180|author=Yu.D. Burago, V.A. Zalgaller, L.D. Kudryavtsev}}
An infinite binary sequence can represent a [[formal language]] (a set of strings) by setting the ''n''&thinsp;th bit of the sequence to 1 if and only if the ''n''&thinsp;th string (in [[shortlex order]]) is in the language. Therefore, the study of [[complexity class]]es, which are sets of languages, may be regarded as studying sets of infinite sequences.
 
An infinite sequence drawn from the alphabet {0, 1, ..., b&minus;1} may also represent a real number expressed in the base-''b'' [[positional number system]]. This equivalence is often used to bring the techniques of [[real analysis]] to bear on complexity classes.
 
== Vectors ==
Sequences over a field may also be viewed as [[Vector (geometric)|vectors]] in a [[vector space]]. Specifically, the set of ''F''-valued sequences (where ''F'' is a [[field (mathematics)|field]]) is a [[function space]] (in fact, a [[product space]]) of ''F''-valued functions over the set of natural numbers.
 
In particular, the term ''[[sequence space]]'' usually refers to a [[linear subspace]] of  the set of all possible infinite sequences with elements in <math>\mathbb{C}</math>.
 
== {{anchor|Doubly infinite|Doubly infinite sequences|Doubly-infinite sequences}} Doubly infinite sequences ==
Normally, the term ''infinite sequence'' refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element (a ''singly infinite sequence''). A ''doubly infinite sequence'' is infinite in both directions—it has neither a first nor a final element. Singly infinite sequences are functions from the natural numbers ('''N''') to some set, whereas doubly infinite sequences are functions from the integers ('''Z''') to some set.
 
One can interpret singly infinite sequences as elements of the [[group ring|semigroup ring]] of the [[natural numbers]] <math>R[\N]</math>, and doubly infinite sequences as elements of the [[group ring]] of the [[integer]]s <math>R[\Z]</math>. This perspective is used in the [[Cauchy product]] of sequences.
 
==Ordinal-indexed sequence==
An [[Order_topology#Ordinal-indexed_sequences|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.
 
== Sequences and automata ==
[[Automata theory|Automata]] or [[finite state machine]]s can typically be thought of as directed graphs, with edges labeled using some specific alphabet, Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence called a ''word'' (or input word). The sequence of states encountered by the automaton when processing a word is called a ''run''. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter.
 
===Types===
*[[±1-sequence]]
*[[Arithmetic progression]]
*[[Cauchy sequence]]
*[[Farey sequence]]
*[[Fibonacci number|Fibonacci sequence]]
*[[Geometric progression]]
*[[Look-and-say sequence]]
*[[Thue–Morse sequence]]
 
===Related concepts===
*[[List (computing)]]
*[[Order_topology#Ordinal-indexed_sequences|Ordinal-indexed sequence]]
*[[Recursion (computer science)]]
*[[Tuple]]
*[[Set theory]]
 
===Operations===
*[[Cauchy product]]
*[[Limit of a sequence]]
 
==See also==
*[[Set (mathematics)]]
*[[Net (topology)]] (a generalization of sequences)
*[[On-Line Encyclopedia of Integer Sequences]]
*[[Permutation]]
*[[Recurrence relation]]
*[[Sequence space]]
 
==References==
{{reflist}}


==External links==
==External links==
*[http://blog.thinkwell.com/2010/07/6th-grade-math-surface-area.html Surface Area Video] at Thinkwell
*[http://oeis.org/ The On-Line Encyclopedia of Integer Sequences]
*[http://www.cs.uwaterloo.ca/journals/JIS/index.html Journal of Integer Sequences] (free)
*{{planetmath reference|id=397|title=Sequence}}


[[Category:Area]]
[[Category:Elementary mathematics]]
[[Category:Sequences and series|*]]


[[es:Área de superficies]]
[[ar:متتالية]]
[[pl:Pole powierzchni]]
[[bg:Редица]]
[[sv:Area]]
[[bs:Niz]]
[[ca:Successió (matemàtiques)]]
[[cs:Posloupnost]]
[[da:Talfølge]]
[[de:Folge (Mathematik)]]
[[et:Jada]]
[[el:Ακολουθία]]
[[es:Sucesión matemática]]
[[eo:Vico]]
[[eu:Segida (matematika)]]
[[fa:دنباله]]
[[fr:Suite (mathématiques)]]
[[gl:Sucesión (matemáticas)]]
[[xal:Даралт]]
[[ko:수열]]
[[hy:Հաջորդականություն (մաթեմատիկական)]]
[[hr:Niz]]
[[io:Sequo]]
[[is:Runa]]
[[it:Successione (matematica)]]
[[he:סדרה]]
[[ka:მიმდევრობა]]
[[kk:Іштізбек]]
[[la:Sequentia (mathematica)]]
[[hu:Sorozat (matematika)]]
[[mk:Низа (математика)]]
[[ml:അനുക്രമം]]
[[ms:Jujukan]]
[[nl:Rij (wiskunde)]]
[[ja:列 (数学)]]
[[no:Følge (matematikk)]]
[[nn:Følgje]]
[[pms:Sequensa]]
[[pl:Ciąg (matematyka)]]
[[pt:Sequência (matemática)]]
[[ro:Șir (matematică)]]
[[ru:Последовательность]]
[[scn:Succissioni (matimatica)]]
[[simple:Sequence]]
[[sk:Postupnosť (matematika)]]
[[sl:Zaporedje]]
[[sr:Низ]]
[[fi:Lukujono]]
[[sv:Följd]]
[[ta:தொடர்வரிசை]]
[[th:ลำดับ]]
[[tr:Dizi (terim)]]
[[uk:Послідовність (математика)]]
[[ur:متوالیہ (ریاضی)]]
[[vi:Dãy (toán học)]]
[[zh:序列]]

Revision as of 03:56, 9 August 2014

I'm Fernando (21) from Seltjarnarnes, Iceland.
I'm learning Norwegian literature at a local college and I'm just about to graduate.
I have a part time job in a the office.

my site; wellness [continue reading this..]

In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.

For example, (M, A, R, Y) is a sequence of letters that differs from (A, R, M, Y), as the ordering matters, and (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 this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. It is, however, bounded.

Examples and notation

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

In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names for convenient referencing. For example a sequence might be written as (a1, a2, a3, … ), or (b0, b1, b2, … ), or (c0, c2, c4, … ), depending on what is useful in the application.

Finite and infinite

A more formal definition of a finite sequence with terms in a set S is a function from {1, 2, ..., n} to S for some n > 0. An infinite sequence in S is a function from {1, 2, ... } to S. For example, the sequence of prime numbers (2,3,5,7,11, … ) is the function 1→2, 2→3, 3→5, 4→7, 5→11, … .

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

A function from all integers into a set is sometimes called a bi-infinite sequence or two-way infinite sequence. An example is the bi-infinite sequence of all even integers ( … , -4, -2, 0, 2, 4, 6, 8… ).

Multiplicative

Let A = (a sequence defined by a function f:{1, 2, 3, ...} → {1, 2, 3, ...}, such that a i = f(i).

The sequence is multiplicative if f(xy) = f(x)f(y) for all x,y such that x and y are coprime.[1]

Types and properties

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.

If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function.

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

If the terms of a sequence are integers, then the sequence is an integer sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence.

If S is endowed with a topology, then it becomes possible to consider convergence of an infinite sequence in S. Such considerations involve the concept of the limit of a sequence.

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

Analysis

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 xn = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N.

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.

Series

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The sum of terms of a sequence is a series. More precisely, if (x1, x2, x3, ...) is a sequence, one may consider the sequence of partial sums (S1, S2, S3, ...), with

Formally, this pair of sequences comprises the series with the terms x1, x2, x3, ..., which is denoted as

If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see series.

Infinite sequences in theoretical computer science

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. Therefore, the study of complexity classes, which are sets of languages, may be regarded as studying sets of infinite sequences.

An infinite sequence drawn from the alphabet {0, 1, ..., b−1} may also represent a real number expressed in the base-b positional number system. This equivalence is often used to bring the techniques of real analysis to bear on complexity classes.

Vectors

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.

In particular, the term sequence space usually refers to a linear subspace of the set of all possible infinite sequences with elements in .

Doubly infinite sequences

Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element (a singly infinite sequence). A doubly infinite sequence is infinite in both directions—it has neither a first nor a final element. Singly infinite sequences are functions from the natural numbers (N) to some set, whereas doubly infinite sequences are functions from the integers (Z) to some set.

One can interpret singly infinite sequences as elements of the semigroup ring of the natural numbers , and doubly infinite sequences as elements of the group ring of the integers . This perspective is used in the Cauchy product of sequences.

Ordinal-indexed sequence

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.

Sequences and automata

Automata or finite state machines can typically be thought of as directed graphs, with edges labeled using some specific alphabet, Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word). The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter.

Types

Related concepts

Operations

See also

References

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External links

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