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[[File:FibonacciBlocks.svg|thumb|220px|right|A tiling with squares whose sides are successive Fibonacci numbers in length.]]
[[File:FibonacciBlocks.svg|thumb|220px|right|A tiling with squares whose sides are successive Fibonacci numbers in length.]]


There are many important integer sequences. The [[prime number]]s are the [[natural numbers]] bigger than 1, that have no [[divisor]]s but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). The study of prime numbers has important applications for [[mathematics]] and specifically [[number theory]].
There are many important integer sequences. The [[prime number]]s are the [[natural numbers]] bigger than 1, that have no [[divisor]]s but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). The study of prime numbers has important applications for [[mathematics]] and specifically [[number theory]].


The [[Fibonacci numbers]] are 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,...).
The [[Fibonacci numbers]] are 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,...).


Other interesting sequences include the [[ban number]]s, whose spellings do not contain a certain letter of the alphabet. For instance, the eban numbers (do not contain 'e') form the sequence (2,4,6,30,32,34,36,40,42,...). Another sequence based on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...).
Other interesting sequences include the [[ban number]]s, whose spellings do not contain a certain letter of the alphabet. For instance, the eban numbers (do not contain 'e') form the sequence (2,4,6,30,32,34,36,40,42,...). Another sequence based on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...).
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:<math>(a_k)_{k=1}^{10}, \qquad a_k = k^2.</math>
:<math>(a_k)_{k=1}^{10}, \qquad a_k = k^2.</math>


Here the subscript {k=1} and superscript 10 together tell us that the elements of this sequence are the ''a<sub>k</sub>'' such that ''k'' = 1, 2, ..., 10.
Here the subscript {k=1} and superscript 10 together tell us that the elements of this sequence are the ''a<sub>k</sub>'' such that ''k'' = 1, 2, ..., 10.


Sequences can be indexed beginning and ending from any integer. The [[infinity]] symbol <big>∞</big> is often used as the superscript to indicate the sequence including all integer ''k''-values starting with a certain one. The sequence of all positive squares is then denoted
Sequences can be indexed beginning and ending from any integer. The [[infinity]] symbol <big>∞</big> is often used as the superscript to indicate the sequence including all integer ''k''-values starting with a certain one. The sequence of all positive squares is then denoted
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In cases where the set of indexing numbers is understood, such as in [[analysis (mathematics)|analysis]], the subscripts and superscripts are often left off. That is, one simply writes ''a<sub>k</sub>'' for an arbitrary sequence. In analysis, k would be understood to run from 1 to ∞. However, sequences are often indexed starting from zero, as in
In cases where the set of indexing numbers is understood, such as in [[analysis (mathematics)|analysis]], the subscripts and superscripts are often left off. That is, one simply writes ''a<sub>k</sub>'' for an arbitrary sequence. In analysis, k would be understood to run from 1 to ∞. However, sequences are often indexed starting from zero, as in
:<math>(a_k)_{k=0}^\infty = ( a_0, a_1, a_2,... ).</math>
:<math>(a_k)_{k=0}^\infty = ( a_0, a_1, a_2,... ).</math>
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 number]]s could be denoted in any of the following ways.
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 number]]s could be denoted in any of the following ways.


* <math>(1,9,25,...)</math>
* <math>(1,9,25,...)</math>
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* <math>((2k-1)^2)_{k=1}^\infty</math>
* <math>((2k-1)^2)_{k=1}^\infty</math>


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]].
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]].


Finally, sequences can more generally be denoted by writing a [[set (mathematics)|set]] inclusion in the subscript, such as in
Finally, sequences can more generally be denoted by writing a [[set (mathematics)|set]] inclusion in the subscript, such as in
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Sequences whose elements are related to the previous elements in a straightforward way are often specified using '''[[Recursive definition|recursion]]'''. This is in contrast to the specification of sequence elements in terms of their position.
Sequences whose elements are related to the previous elements in a straightforward way are often specified using '''[[Recursive definition|recursion]]'''. This is in contrast to the specification of sequence elements in terms of their position.


To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones before it. In addition, enough initial elements must be specified so that new elements of the sequence can be specified by the rule. The principle of [[mathematical induction]] can be used to [[proof (mathematics)|prove]] that a sequence is [[well-defined]], which is to say that that every element of the sequence is specified at least once and has a single, unambiguous value. Induction can also be used to prove properties about a sequence, especially for sequences whose most natural specification is by recursion.
To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones before it. In addition, enough initial elements must be specified so that new elements of the sequence can be specified by the rule. The principle of [[mathematical induction]] can be used to [[proof (mathematics)|prove]] that a sequence is [[well-defined]], which is to say that that every element of the sequence is specified at least once and has a single, unambiguous value. Induction can also be used to prove properties about a sequence, especially for sequences whose most natural specification is by recursion.


The [[Fibonacci sequence]] can be defined using a recursive rule along with two initial elements. The rule is that each element is the sum of the previous two elements, and the first two elements are 0 and 1.
The [[Fibonacci sequence]] can be defined using a recursive rule along with two initial elements. The rule is that each element is the sum of the previous two elements, and the first two elements are 0 and 1.
:<math>a_n = a_{n-1} + a_{n-2}</math>, {{pad|4em}} with {{pad|2em}} ''a''<sub>0</sub> = 0 {{pad|.5em}}and{{pad|.5em}} ''a''<sub>1</sub> = 1.
:<math>a_n = a_{n-1} + a_{n-2}</math>, {{pad|4em}} with {{pad|2em}} ''a''<sub>0</sub> = 0 {{pad|.5em}}and{{pad|.5em}} ''a''<sub>1</sub> = 1.
The first ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence that is defined recursively is [http://oeis.org/A005132 Recaman's sequence], considered at the beginning of this section. We can define Recaman's sequence by
The first ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence that is defined recursively is [http://oeis.org/A005132 Recaman's sequence], considered at the beginning of this section. We can define Recaman's sequence by
:''a''<sub>0</sub> = 0 {{pad|2em}}and{{pad|2em}} ''a<sub>n</sub>'' = ''a<sub>n''−1</sub>−''n'' {{pad|1em}} if the result is positive and not already in the list. Otherwise, {{pad|1em}} ''a<sub>n</sub>'' = ''a<sub>n''−1</sub>+''n'' .
:''a''<sub>0</sub> = 0 {{pad|2em}}and{{pad|2em}} ''a<sub>n</sub>'' = ''a<sub>n''−1</sub>−''n'' {{pad|1em}} if the result is positive and not already in the list. Otherwise, {{pad|1em}} ''a<sub>n</sub>'' = ''a<sub>n''−1</sub>+''n'' .


Not all sequences can be specified by a rule in the form of an equation, recursive or not, and some can be quite complicated. For example, the sequence of [[prime number]]s is the set of prime numbers in their natural order. This gives the sequence (2,3,5,7,11,13,17,...).
Not all sequences can be specified by a rule in the form of an equation, recursive or not, and some can be quite complicated. For example, the sequence of [[prime number]]s is the set of prime numbers in their natural order. This gives the sequence (2,3,5,7,11,13,17,...).
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:<math>\lim_{n\to\infty} a_n = L.</math>
:<math>\lim_{n\to\infty} a_n = L.</math>


More precisely, the sequence converges if there exists a limit L such that the remaining a<sub>n</sub>'s are arbitrarily close to L for some n large enough.
More precisely, the sequence converges if there exists a limit L such that the remaining a<sub>n</sub>'s are arbitrarily close to L for some n large enough.


If a sequence converges to some limit, then it is '''convergent'''; otherwise it is '''divergent'''.
If a sequence converges to some limit, then it is '''convergent'''; otherwise it is '''divergent'''.
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If ''a<sub>n</sub>'' gets arbitrarily large as ''n'' → ∞ we write
If ''a<sub>n</sub>'' gets arbitrarily large as ''n'' → ∞ we write
:<math>\lim_{n\to\infty}a_n = \infty.</math>
:<math>\lim_{n\to\infty}a_n = \infty.</math>
In this case the sequence (''a<sub>n</sub>'') ''diverges'', or that it converges to infinity.
In this case the sequence (''a<sub>n</sub>'') ''diverges'', or that it converges to infinity.


If ''a<sub>n</sub>'' becomes arbitrarily "small" negative numbers (large in magnitude) as ''n'' → ∞ we write
If ''a<sub>n</sub>'' becomes arbitrarily "small" negative numbers (large in magnitude) as ''n'' → ∞ we write
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* [http://oeis.org/ The On-Line Encyclopedia of Integer Sequences]
* [http://oeis.org/ The On-Line Encyclopedia of Integer Sequences]
* [http://www.cs.uwaterloo.ca/journals/JIS/index.html Journal of Integer Sequences] (free)
* [http://www.cs.uwaterloo.ca/journals/JIS/index.html Journal of Integer Sequences] (free)
* {{en}} {{Templat:PlanetMath|sequence|sequence (ID: barisan)}}
* {{en}} {{PlanetMath|sequence|sequence (ID: barisan)}}


<!--[[Category:Elementary mathematics]]
<!--[[Category:Elementary mathematics]]
[[Category:Sequences and series]]-->
[[Category:Sequences and series]]-->


[[Kategori:Matematika]]
[[Kategori:Matematika]]{{Authority control}}

Revisi per 6 Juli 2021 05.35

Barisan dalam matematika, adalah suatu daftar tertata. Sebagaimana suatu himpunan, urutan memuat "anggota" atau "elemen" (juga disebut "suku" atau "istilah"). Jumlah elemen tertata (kemungkinan tak terhingga) disebut panjang urutan. Berbeda dengan himpunan, penataan urutan sangat penting dan elemen-elemen yang tepat sama dapat muncul berulang kali pada posisi berbeda dalam urutan itu. Lebih tepatnya, suatu urutan dapat didefinisikan sebagai suatu fungsi di mana ranah (atau domain) darinya merupakan suatu himpunan countable totally ordered, sepertu bilangan asli.

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Referensi

Pranala luar

  • Definisi kamus barisan di Wikikamus
  • Hazewinkel, Michiel, ed. (2001) [1994], "Sequence", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 
  • The On-Line Encyclopedia of Integer Sequences
  • Journal of Integer Sequences (free)
  • (Inggris) sequence (ID: barisan) di PlanetMath.
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