What is Sequences: Definition and 586 Discussions

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 whose domain is either the set of the natural numbers (for infinite sequences), or the set of the first n natural numbers (for a sequence of finite length n). Sequences are one type of indexed families as an indexed family is defined as a function which domain is called the index set, and the elements of the index set are the indices for the elements of the function image.
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




a

n




{\displaystyle a_{n}}
,




b

n




{\displaystyle b_{n}}
and




c

n




{\displaystyle c_{n}}
, where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence



F


{\displaystyle F}
is generally denoted as




F

n




{\displaystyle F_{n}}
.

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.

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  1. B

    Limit proof on monotonic sequences

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  2. E

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  3. G

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  4. D

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  5. H

    Two monotone sequences proof: Prove lim(an)<=lim(bn)

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  6. R

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  7. K

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  8. R

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  9. M

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  11. M

    Sequences and convergence in the standard topology

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  12. R

    Understanding Monotonic Sequences: Simplification and Frustration

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  13. I

    Why Continuous Functions Don't Preserve Cauchy Sequences

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  14. J

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  15. T

    Sequences and Series Problem. Help Pleeaassee

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  16. Q

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  17. Loren Booda

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  18. L

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  19. A

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  20. H

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  21. C

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  22. B

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  23. matqkks

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  24. 8

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  25. Z

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  29. Z

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  30. C

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  31. T

    Sequences and continuous functions

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  32. E

    Product of 2 Increasing Sequences Not Necessarily Increasing

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  33. J

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  34. J

    Applying Cantor's diagonalization technique to sequences of functions

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  35. J

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  37. E

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  38. G

    How Does the Behavior of k^n Change with Different Ranges of k?

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  39. R

    Cauchy sequences and continuity versus uniform continuity

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  40. R

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  41. G

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  48. F

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  49. P

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  50. R

    Set of all rational sequences countable?

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