Limit of a sequence

Last updated
The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e.
2
p
r
.
{\displaystyle 2\pi r.}
The corresponding sequence for inscribed polygons has the same limit. Archimedes pi.svg
The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. The corresponding sequence for inscribed polygons has the same limit.
nn sin(1/n)
10.841471
20.958851
...
100.998334
...
1000.999983

As the positive integer becomes larger and larger, the value becomes arbitrarily close to We say that "the limit of the sequence equals "

Contents

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). [1] [2] If such a limit exists, the sequence is called convergent. [3] A sequence that does not converge is said to be divergent. [4] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. [2]

Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.

History

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes.

Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series.

Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite series (written in 1671, published in English translation in 1736, Latin original published much later) and Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to his Optiks). In the latter work, Newton considers the binomial expansion of (x + o)n, which he then linearizes by taking the limit as o tends to 0.

In the 18th century, mathematicians such as Euler succeeded in summing some divergent series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated the conditions under which a series converged to a limit.

The modern definition of a limit (for any ε there exists an index N so that ...) was given by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, which was little noticed at the time), and by Karl Weierstrass in the 1870s.

Real numbers

The plot of a convergent sequence {an} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases. Converging Sequence example.svg
The plot of a convergent sequence {an} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases.

In the real numbers, a number is the limit of the sequence if the numbers in the sequence become closer and closer to —and not to any other number.

Examples

Formal definition

We call the limit of the sequence if the following condition holds:

In other words, for every measure of closeness the sequence's terms are eventually that close to the limit. The sequence is said to converge to or tend to the limit written or

Symbolically, this is:

If a sequence converges to some limit then it is convergent and is the only limit; otherwise is divergent. A sequence that has zero as its limit is sometimes called a null sequence.

Illustration

Properties

Limits of sequences behave well with respect to the usual arithmetic operations. If and then and, if neither b nor any is zero, [5]

For any continuous function f, if then In fact, any real-valued function f is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity).

Some other important properties of limits of real sequences include the following (provided, in each equation below, that the limits on the right exist).

These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition. For example. once it is proven that it becomes easy to show—using the properties above—that (assuming that ).

Infinite limits

A sequence is said to tend to infinity, written or if for every K, there is an N such that for every ; that is, the sequence terms are eventually larger than any fixed K.

Similarly, if for every K, there is an N such that for every If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence provides one such example.

Metric spaces

Definition

A point of the metric space is the limit of the sequence if for all there is an such that, for every This coincides with the definition given for real numbers when and

Properties

For any continuous function f, if then In fact, a function f is continuous if and only if it preserves the limits of sequences.

Limits of sequences are unique when they exist, as distinct points are separated by some positive distance, so for less than half this distance, sequence terms cannot be within a distance of both points.

Topological spaces

Definition

A point of the topological space is a limit or limit point [7] [8] of the sequence if for every neighbourhood of there exists some such that for every [9] This coincides with the definition given for metric spaces, if is a metric space and is the topology generated by

A limit of a sequence of points in a topological space is a special case of a limit of a function: the domain is in the space with the induced topology of the affinely extended real number system, the range is and the function argument tends to which in this space is a limit point of

Properties

In a Hausdorff space, limits of sequences are unique whenever they exist. Note that this need not be the case in non-Hausdorff spaces; in particular, if two points and are topologically indistinguishable, then any sequence that converges to must converge to and vice versa.

Cauchy sequences

The plot of a Cauchy sequence (xn), shown in blue, as
x
n
{\displaystyle x_{n}}
versus n. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence converges to some limit. Cauchy sequence illustration.svg
The plot of a Cauchy sequence (xn), shown in blue, as versus n. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence converges to some limit.

A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is the Cauchy criterion for convergence of sequences: a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains true in other complete metric spaces.

Definition in hyperreal numbers

The definition of the limit using the hyperreal numbers formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a real sequence tends to L if for every infinite hypernatural H, the term is infinitely close to L (i.e., the difference is infinitesimal). Equivalently, L is the standard part of

Thus, the limit can be defined by the formula

where the limit exists if and only if the righthand side is independent of the choice of an infinite H.

See also

Notes

    1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-18.
    2. 1 2 Courant (1961), p. 29.
    3. Weisstein, Eric W. "Convergent Sequence". mathworld.wolfram.com. Retrieved 2020-08-18.
    4. Courant (1961), p. 39.
    5. 1 2 3 4 5 6 7 8 "Limits of Sequences | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-18.
    6. Weisstein, Eric W. "Limit". mathworld.wolfram.com. Retrieved 2020-08-18.
    7. Dugundji 1966, pp. 209-210.
    8. Császár 1978, p. 61.
    9. Zeidler, Eberhard (1995). Applied functional analysis : main principles and their applications (1 ed.). New York: Springer-Verlag. p. 29. ISBN   978-0-387-94422-7.

    Proofs

    1. Proof: choose For every
    2. Proof: choose (the floor function). For every

    Related Research Articles

    In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

    In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

    Real analysis Mathematics of real numbers and real functions

    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 functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

    Sequence Finite or infinite ordered list of elements

    In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements 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 from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.

    Limit inferior and limit superior Bounds of a sequence

    In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. They can be thought of in a similar fashion for a function. For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

    In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set if, given any arbitrarily small positive number , a number can be found such that each of the functions differ from by no more than at every pointin. Described in an informal way, if converges to uniformly, then the rate at which approaches is "uniform" throughout its domain in the following sense: in order to guarantee that falls within a certain distance of , we do not need to know the value of in question — there can be found a single value of independent of , such that choosing will ensure that is within of for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find so that, for that particular, falls within of whenever .

    In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if

    In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

    In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

    In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.

    Squeeze theorem On calculating limits by bounding a function between two other functions

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

    Integral test for convergence

    In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

    The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

    In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

    In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted

    In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian physicist and geometer, who published independent proofs respectively in 1910 and 1911.

    Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.

    In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in Lp in terms of convergence in measure and a condition related to uniform integrability.

    In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergent sequences into convergent sequences: if xnx then g(xn) → g(x). The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {xn} with a sequence of random variables {Xn}, and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables.

    In mathematics, a limit is the value that a function approaches as the input approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

    References