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. P

    I How does the ratio test fail and the root test succeed here?

    The series that is given is $$\frac12+\frac13+\left(\frac12\right)^2+\left(\frac13\right)^2+\left(\frac12\right)^3+\left(\frac13\right)^3+\ldots.$$ Now, it's easy to see these are two separate geometric series, however, Spivak claims the ratio test fails because the ratio of successive terms...
  2. G

    Calculating Shannon Entropy of DNA Sequences

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  3. A

    Probability involving Gaussian random sequences

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  4. mcastillo356

    Why does this function make it easy to prove continuity with sequences?

    I've been given the proof, but don't understand; to calculate the limit of ##f## when ##x## tends to zero it's enough to see that if ##\{x_n\}_{n=1}^\infty## is a sequence that tends to ##0##, then...
  5. M

    B Golden Ratio in Collatz-like sequences

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  6. A

    Sinusoidal sequences with random phases

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  7. Euge

    POTW Infinite Sequences of Sines

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  8. A

    Trust Fund problem using series and sequences

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  9. Vividly

    B Understanding about Sequences and Series

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  10. chwala

    Bounded and monotonic sequences - Convergence

    I would like some clarity on the highlighted part. My question is, consider the the attached example ##(c)##, This sequence converges ( by using L'Hopital's rule)...now my question is, the sequence is indicated on text as not being monotonic...very clear. Does it imply that if a sequence is not...
  11. A

    I Limits of Two Ordinal Sequences

    Let ##\omega_1## be the first uncountable ordinal such that ##x## is an element of ##\omega_1## if and only if it is either a finite ordinal or there exists a bijection from ##x## onto ##\omega##. I want to define a matrix such that the matrix contains each element of ##\omega_1## only once. To...
  12. BerriesAndCream

    Calculating nth Term of Sequences: What Now?

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  13. B

    Proof for Cauchy sequences

    I've started by writing down the definitions, so we have $$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$ $$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w...
  14. C

    MHB Upper Bound of Sets and Sequences: Analyzing Logic

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  15. yucheng

    Understanding the Use of Min in Cauchy Sequences

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  16. yucheng

    Proof that two equivalent sequences are both Cauchy sequences

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  17. A

    DNA sequencing and restoring malformed sequences

    I was just reading about DNA sequencing. In my view, DNA can be modeled into an ordered sequence of nucleobases, as if the two strands were joined into a single strand (just like in RNA). The first half of the sequence models the first strand. The four nucleobases are numbered from 0 to 3...
  18. Purpleshinyrock

    How to Solve Arithmetic Sequence Problems

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  19. T

    I Limits of functions and sequences

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  20. penroseandpaper

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  21. I

    B Weird stuff on infinite numerical sequences in a Soviet book

    The book is Calculus: Basic Concepts for High School on the first page you are given the following sequence: 1, -1, 1/3, -1/3, 1/5, -1/5, 1/7, -1/7, ... several pages later the rule is given: in the second rule, for the first term in the sequence, the coefficient of one of the terms is 1/0...
  22. S

    I Convergence of sequences of functions with differing domains?

    Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let ##F_n(x)## be defined by ##F_n(x) = 1 + h## where ##h...
  23. WMDhamnekar

    MHB Solve Math Problem: Mixing Milk & Water to Get 50% Milk

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  24. M

    MHB Topic of presentation: Elementary Geometry vs Fibonacci & its sequences

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  25. J

    MHB Number patterns and sequences - Tn Term

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  26. sergey_le

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  27. M

    MHB Check convergence of sequences

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  28. Math Amateur

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  29. Math Amateur

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  30. F

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  31. TytoAlba95

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  32. Math Amateur

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  33. Math Amateur

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  34. Math Amateur

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  35. Math Amateur

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  36. V

    MHB GCD of Two Specified Sequences of Numbers: Conditions for Equality

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  37. J

    MHB How to exclude combinations for defined sequences?

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  38. WMDhamnekar

    MHB Check Martingale Sequences from i.i.d. Variables | Stats SE

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  39. F

    I Fisher matrix - equivalence or not between sequences

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  40. L

    A Same open sets + same bounded sets => same Cauchy sequences?

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  41. Entertainment Unit

    Find bounding numbers for two interrelated sequences

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  42. M

    B Recursive sequences - notation question

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  43. Mr Davis 97

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  44. S

    I Sequences for infinitely nested radicals

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  45. Mr Davis 97

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  46. Mr Davis 97

    Convergent sequences might not have max or min

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  47. J

    MHB Real Analysis, Sequences in relation to Geometric Series and their sums

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  48. Robin04

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  49. M

    I Question regarding a sequence proof from a book

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  50. T

    Distance of a point from a compact set in ##\Bbb{R}##

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