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Topic: Limit of a sequence

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Cauchy sequence

Complete space

Infinite series

Totally bounded

Simplex

Real number

Normal space

Compact space

Extended real number line

Prime number

Topological space

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	Limit of a sequence - Wikipedia, the free encyclopedia
		The limit of a sequence is one of the oldest concepts in mathematical analysis.
		A point L is the limit of the sequence if for any prescribed nearness, all but a finite number of points in the sequence are that near to L.
		A bounded monotonic sequence of real numbers is necessarily convergent: this is known as the monotone convergence theorem.
	en.wikipedia.org /wiki/Limit_of_a_sequence (882 words)

	Limit (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either gets "close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements, as their index increases indefinitely.
		The concept of the "limit of a function" is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.
		The limit of a sequence and the limit of a function are closely related.
	en.wikipedia.org /wiki/Limit_(mathematics) (762 words)

	HighBeam Encyclopedia â“ Free Online Encyclopedia for Reference, Research, Facts
		LIMIT [limit] in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity.
		For example, the terms of the sequence 1/2, 1/4, 1/8, 1/16, … are obviously getting smaller and smaller; since, if enough terms are taken, one can make the last term as small, i.e., as close to zero, as one pleases, the limit of this sequence is said to be zero.
		Similarly, the sequence 3, 5, 3 1/2, 4 1/2, 3 3/4, 4 1/4, 3 7/8, 4 1/8, … is seen to approach 4 as a limit.
	www.encyclopedia.com /printable.aspx?id=1E1:limit (339 words)

	AllRefer.com - limit (Mathematics) - Encyclopedia
		limit, in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity.
		For example, the terms of the sequence 1/2, 1/4, 1/8, 1/16, … are obviously getting smaller and smaller; since, if enough terms are taken, one can make the last term as small, i.e., as close to zero, as one pleases, the limit of this sequence is said to be zero.
		Similarly, the sequence 3, 5, 3 1/2, 4 1/2, 3 3/4, 4 1/4, 3 7/8, 4 1/8, … is seen to approach 4 as a limit.
	reference.allrefer.com /encyclopedia/L/limit.html (382 words)

	Limit superior and limit inferior
		In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence.
		The value of this limit inferior is conjectured to be 2 - this is the Twin Prime Conjecture - but as yet has not even been proved finite.
		The power set P(X) of a set X is a complete lattice, and it is sometimes useful to consider limits superior and inferior of sequences in P(X), that is, sequences of subsets of X.
	www.ebroadcast.com.au /lookup/encyclopedia/li/Lim_sup.html (340 words)

	Limit (mathematics) : Limit
		The mathematical concept of limit is used to describe the behavior of a function as its argument get "close" to some point (or attempts to get close to infinity), or the behavior of the elements of a sequence as their index approaches infinity.
		Limits are more interesting when they are unreachable; for example: if f(x) = (x³ - 1) / (x - 1) then, x cannot equal 1 (as that would result in division by zero); however, f(x) does approach some number c (as x approaches 1).
		The rest of this article presents the concept of limit in increasing generality, starting with sequences and functions of real numbers, then metric spaces, and culminating in the most general concept, limits of nets on topological spaces.
	www.wordlookup.net /li/limit.html (2692 words)

	The meaning of a limit - An approach to calculus
		The most important limit -- the limit on which calculus hinges -- is called the derivative; all the other limits studied in Calculus I are logical fun and games, never to be heard from again.
		The limit of a sum is equal to the sum of the limits.
		The limit of a quotient is equal to the quotient of the limits,
	www.themathpage.com /aCalc/limits.htm (2033 words)

	Limits...Taback's Child's Concept of Limit
		The concept of limit is one of the most important in mathematics.
		The first four squares of a sequence of diverging squares are presented to the child.
		Since the domain of a sequence is tacitly understood to be the set of natural numbers, the sequence is often stated simply in terms of its range S
	www.unf.edu /~tbratina/mathed/limits/divergesquaresteacherpage.htm (523 words)

	Poster Project, What Is Scientific Truth Poster, Zeno's Paradoxes (Site not responding. Last check: 2007-10-31)
		You can read more about limits on the Analysis page, but one thing to bear in mind is that here I am talking about the limit of a sequence (as opposed to the limit of a function, as I am on the Analysis page).
		By the limit of a sequence I mean that number which the terms in the sequence get closer and closer to the further along the sequence you go.
		Not all sequences have limits, and even if a sequence has a limit it is important to realize that the numbers in the sequence may never get to that limit.
	www.math.sunysb.edu /posterproject/www/materials/truth/theorem2.html (955 words)

	Sequences
		Sequences are, basically, countably many numbers arranged in a sequence that may or may not exhibit certain patterns.
		It is often easier to simply look at a sequence as a 'string' of numbers that may or may not exhibit a certain pattern.
		Here is a very useful theorem to establish convergence of a given sequence (without, however, revealing the limit of the sequence): First, we have to apply our concepts of supremum and infimum to sequences:
	pirate.shu.edu /projects/reals/numseq/sequence.html (720 words)

	Real Numbers as Equivalence Classes of Cauchy Convergent Sequences
		The conceptual difficulty in defining real numbers as convergent sequences is that there is no entities to be taken as the limits of convergent sequences.
		Thus an unbounded sequence cannot be convergent and a convergent sequence cannot be unbounded.
		Two convergent sequences are equivalent; i.e., belong to the same equivalence class, is their difference is in the equivalence class of zero.
	www2.sjsu.edu /faculty/watkins/cauchy.htm (1120 words)

	SEQUENCES
		The sequence may be finite or infinite, reveal a pattern or appear entirely random.
		First we should note that in mathematical circles the formal definition of a sequence is commonly cast in the form of a function.
		The Length of the Hailstone sequence is the number of terms in the sequence until the subsequence 4, 2, 1 appears.
	isolatium.uhh.hawaii.edu /m206L/lab9/sequences/sequences.htm (903 words)

	Search Results for Sequence*
		Then the "curious property" is that each member of the sequence is equal to the rational whose numerator is the sum of the numerators of the fractions on either side, and whose denominator is the sum of the denominators of the fractions on either side.
		In 1907 Ernst Fischer studied orthonormal sequences of functions and gave necessary and sufficient conditions for a sequence of constants to be the Fourier coefficients of a square integrable function.
		Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Sequence*&CONTEXT=1 (5617 words)

	The Limit of a Sequence
		The limit point p of the sequence s is quite different from the limit point of the set s.
		The point p is a limit point of the set r iff p is the limit of a convergent sequence taken from r.
		The point p is a cluster point of the sequence r iff p is the limit of a convergent subsequence of r.
	www.mathreference.com /top,clust.html (1056 words)

	Limit of a Sequence (Site not responding. Last check: 2007-10-31)
		Convergence of a sequence to a limit L can now be expressed in terms of decimal numbers or significant digits: For any whole number k, there is a whole number N, such that all terms in the sequence after the first N agree with the limit L to k decimal places.
		The leftmost limit point has a decimal expansion computed to k decimals by covering the interval with nicely aligned subintervals of length 10**(-k) and locating the leftmost one with infinitely points (or concluding that such an interval most exist.
		Thus a sequence of nested intervals with left end points converging to the "lim inf" (technical expression) of the set is obtained.
	whyslopes.com /Calculus-Introduction/limit_of_a_sequence.html (1700 words)

	All Elementary Mathematics - Study Guide - Principles of analysis - Sequences. Limits of natural sequences. Some ...
		Note, that it is not always possible to give the numerical sequence by a general term formula; sometimes a sequence is given by description of its terms (see below the last example).
		This definition means, that a is a limit of a numerical sequence, if its general term approaches unrestrictedly to a at increasing n.
		Each monotone and bounded sequence has a limit (this theorem is used in a high school without a proof).
	www.bymath.com /studyguide/ana/sec/ana1.htm (357 words)

	PostgreSQL: Documentation: Manuals: PostgreSQL 8.0: CREATE SEQUENCE (Site not responding. Last check: 2007-10-31)
		Temporary sequences exist in a special schema, so a schema name may not be given when creating a temporary sequence.
		The sequence name must be distinct from the name of any other sequence, table, index, or view in the same schema.
		Furthermore, although multiple sessions are guaranteed to allocate distinct sequence values, the values may be generated out of sequence when all the sessions are considered.
	www.postgresql.org /docs/8.0/interactive/sql-createsequence.html (727 words)

	PostgreSQL: Documentation: Manuals: PostgreSQL 7.3: CREATE SEQUENCE
		The defaults are 1 and -2^63-1 for ascending and descending sequences, respectively.
		TEMP sequences exist in a special schema, so a schema name may not be given when creating a TEMP sequence.
		Furthermore, although multiple backends are guaranteed to allocate distinct sequence values, the values may be generated out of sequence when all the backends are considered.
	www.postgresql.org /docs/7.3/interactive/sql-createsequence.html (770 words)

	Cauchy Sequences (Site not responding. Last check: 2007-10-31)
		What is slightly annoying for the mathematician (in theory and in praxis) is that we refer to the limit of a sequence in the definition of a convergent sequence when that limit may not be known at all.
		In fact, more often then not it is quite hard to determine the actual limit of a sequence.
		Thus, by considering Cauchy sequences instead of convergent sequences we do not need to refer to the unknown limit of a sequence, and in effect both concepts are the same.
	pirate.shu.edu /projects/reals/numseq/causeq.html (288 words)

	Sequences
		To evaluate such a limit, realize that each sequence is associated with a continuous function.
		To evaluate the limit of a sequence we simply find the limit of the function associated with this sequence.
		Then to evaluate this limit we must associate the positive terms with one function and the negative terms with that functions negative.
	www-math.cudenver.edu /~rrosterm/review_2/node4.html (152 words)

	Limits of Sequences (Site not responding. Last check: 2007-10-31)
		Notice that this lemma only goes one way: having the function have L as limit is much stronger than having the sequence have L as limit.
		It is quite possible for the sequence to converge but to have no limit for the function.
		Monotone means that the sequence either always increases or always decreases rather than mixing them up and sometimes increasing and sometimes decreasing.
	www.iwu.edu /~lstout/sequences/node2.html (422 words)

	limit - HighBeam Encyclopedia
		Term limit issue split, with polls vs. council: Voters have the option Aug. 8 to overturn cutoff on leaders' time in office or maintain status quo.
		Height limit proposed: 40 feet would be Folly maximum.
		Loss limit goes back on table: Effect on gamblers worries opponents
	www.encyclopedia.com /doc/1E1-limit.html (551 words)

	Texts and tools for the Online Encylopedia of Integer Sequences
		Almost 10,000 sequences (May 2003) are not marked as dead, dupe, full, hard, obsc, unkn, or more, where the third line of terms is empty and no term is longer than 30 digits.
		The special case of a sequence without signed terms marked as sign but not nonn is only reported, because the signed terms may start later, see A057752 resp.
		Various Shell sort OEIS sequences are tested: Shell's A003462, Knuth's A033622, and A055876, A055875, A036562, A036564, A036569.
	www.xyzzy.claranet.de /eis.htm (890 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		In mathematics sequence is defined as a function whose domain is a set of positive integers.
		Properties of limits of sequences The properties of limits of sequences are similar to the properties of limits of functions.
		Sequence is bounded and monotonic; it is convergent sequence. Example: Determine whether the sequence is monotonic.
	www.iitap.iastate.edu /arabiccalc/hao/chap8.doc (1285 words)

	Limit points and closed sets in metric spaces
		The points 0 and 1 are both limit points of the interval (0, 1).
		Any point on the boundary of the circle is a limit of a sequence of points inside the circle.
		Every real number can be approximated arbitrarily closely by a sequence of rationals (by truncating the decimal expansion, say).
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L9.html (463 words)

	On the limit of a sequence
		The object of this article is to examine the sequence
		We prove that the sequence is bounded, strictly monotonously decreasing, and
		Finally, a modification and a generalization of (an) will be mentioned, and the sketch of a second analytical proof for the original limit will be given.
	www.univie.ac.at /EMIS/journals/AMAPN/vol15/voros.abstract.html (56 words)

	Jiskha Homework Help - Mathematics: Limit of a Sequence
		The notion of limit of a sequence is very natural.
		In fact, you are right and even more than that; this sequence is completely chaotic (see the Figure below for the first 50 terms of the sequence).
		By the way, this sequence is used as a discrete mathematical model for population dynamics (called the discrete logistic model).
	www.jiskha.com /mathematics/calculus/limit_of_a_sequence.html (726 words)

	Question Corner -- Limit of the Sequence a(n) = cos(a(n-1))
		The number this sequence converges to is the unique solution to the equation x = cos x.
		The reason the sequence converges to this number z is as follows.
		Now, regardless of what the first term of the sequence is, the second and all subsequent terms are the cosine of something and hence are between -1 and 1.
	www.math.toronto.edu /mathnet/plain/questionCorner/transseq.html (768 words)

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