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Limit of a sequence

Index Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1. [1]

51 relations: Antiphon (orator), Archimedes, Arithmetic–geometric mean, Bernard Bolzano, Carl Friedrich Gauss, Complete metric space, Continuous function, Decimal representation, Democritus, E (mathematical constant), Eudoxus of Cnidus, Extended real number line, Floor and ceiling functions, Frank Morley, Function (mathematics), Geometric series, Hausdorff space, Hypergeometric function, Hyperinteger, Hyperreal number, Induced topology, Infinitesimal, Integer, Isaac Newton, James Harkness, Joseph-Louis Lagrange, Karl Weierstrass, Leonhard Euler, Leucippus, Limit of a function, Limit point, Mathematical analysis, Mathematician, Mathematics, Method of exhaustion, Metric space, Modes of convergence, Natural number, Neighbourhood (mathematics), Net (mathematics), Real analysis, Real number, Richard Courant, Sequence, Shift rule, Squeeze theorem, Standard part function, Topological indistinguishability, Topological space, Zeno of Elea, ..., Zeno's paradoxes. Expand index (1 more) »

Antiphon (orator)

Antiphon of Rhamnus (Ἀντιφῶν ὁ Ῥαμνούσιος) (480–411 BC) was the earliest of the ten Attic orators, and an important figure in fifth-century Athenian political and intellectual life.

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Archimedes

Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

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Arithmetic–geometric mean

In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers and is defined as follows: Call and and: \end Then define the two interdependent sequences and as \end where the square root takes the principal value.

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Bernard Bolzano

Bernard Bolzano (born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views.

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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Decimal representation

A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum where a0 is a nonnegative integer, and a1, a2,...

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Democritus

Democritus (Δημόκριτος, Dēmókritos, meaning "chosen of the people") was an Ancient Greek pre-Socratic philosopher primarily remembered today for his formulation of an atomic theory of the universe.

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E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

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Eudoxus of Cnidus

Eudoxus of Cnidus (Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato.

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Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: and (read as positive infinity and negative infinity respectively).

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Floor and ceiling functions

In mathematics and computer science, the floor function is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x, denoted \operatorname(x).

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Frank Morley

Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive terms.

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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

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Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

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Hyperinteger

In non-standard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part.

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Hyperreal number

The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.

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Induced topology

In topology and related areas of mathematics, an induced topology on a topological space is a topology which makes the inducing function continuous from/to this topological space.

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Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

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James Harkness

James Harkness (1864–1923) was a Canadian mathematician, born in Derby, England, and educated at Trinity College, Cambridge.

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Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

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Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

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Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

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Leucippus

Leucippus (Λεύκιππος, Leúkippos; fl. 5th cent. BCE) is reported in some ancient sources to have been a philosopher who was the earliest Greek to develop the theory of atomism—the idea that everything is composed entirely of various imperishable, indivisible elements called atoms.

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Limit of a function

Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

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Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Method of exhaustion

The method of exhaustion (methodus exhaustionibus, or méthode des anciens) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Modes of convergence

In mathematics, there are many senses in which a sequence or a series is said to be convergent.

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Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

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Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

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Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.

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Real analysis

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-valued functions.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Richard Courant

Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician.

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Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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Shift rule

The shift rule is a mathematical rule for sequences and series.

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Squeeze theorem

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function.

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Standard part function

In non-standard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers.

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Topological indistinguishability

In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Zeno of Elea

Zeno of Elea (Ζήνων ὁ Ἐλεάτης) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides.

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Zeno's paradoxes

Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion.

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Redirects here:

Convergent sequence, Divergent sequence, Null sequence, Topological convergence.

References

[1] https://en.wikipedia.org/wiki/Limit_of_a_sequence

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