Definitions

Limit_of_a_sequence

Limit of a sequence

The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit.

Intuitively, suppose we have a sequence of points (i.e. an infinite set of points labelled using the natural numbers) in some sort of mathematical object (for example the real numbers or a vector space) which has a concept of nearness (such as "all points within a given distance of a fixed point"). 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. This may be visualised as a set of spheres of size decreasing to zero, all with the same centre L, and for any one of these spheres, only a finite number of points in the sequence being outside the sphere.

Formal definition

  • For a sequence of real numbers {x_n|nin mathbb{N}};

A real number L is said to be the limit of the sequence xn, written

lim_{n to infty} x_n=L,

if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have |xnL| < ε.

An element Lin M is said to be the the limit of the sequence, written

lim_{n to infty} x_n = L,

if and only if for every real number ε >  there exists a natural number N such that for every n > N, we have d(xn,L) < ε.

  • As a generalization of this, for a sequence of points {x_n|nin mathbb{N}}; in a topological space T:

An element Lin T; is said to be a limit of this sequence, written

lim_{n to infty} x_n=L,

if and only if for every neighborhood S of L there is a natural number N such that x_nin S; for all n>N.;

If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is divergent (see also oscillation).

A null sequence is a sequence that converges to 0.

Comments

The definition means that eventually all elements of the sequence get as close as we want to the limit. (The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence).

A sequence of real numbers may tend to +infty or -infty, compare . Even though this can be written in the form

lim_{n to infty} x_n = infty and lim_{n to infty} x_n = -infty

such a sequence is called divergent, unless we explicitly consider it a sequence in the affinely extended real number system or (in the first case only) the real projective line. In the latter cases the sequence has a limit (in the space itself), so could be called convergent, but when using this term here, care should be taken that this does not cause confusion.

Also, a sequence may, in a general topological space, have several different limits, but a convergent sequence has a unique limit if T is a Hausdorff space, for example the (extended) real line, the complex plane, their subsets (R, Q, Z...) and Cartesian products (Rn...).

The limit of a sequence of points {x_n|nin mathbb{N}}; in a topological space T is a special case of the : the domain is mathbb{N} in the space mathbb{N} cup lbrace +infty rbrace with the induced topology of the affinely extended real number system, the range is T, and the function argument n tends to +∞, which in this space is a limit point of mathbb{N}.

Examples

  • The sequence 1, -1, 1, -1, 1, ... is divergent.
  • The sequence 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, ... converges with limit 1. This is an example of an infinite series.
  • If a is a real number with absolute value |a| < 1, then the sequence an has limit 0. If 0 < a ≤ 1, then the sequence a1/n has limit 1.

Also:

lim_{ntoinfty} frac{1}{n^p} = 0 hbox{ if } p > 0

lim_{ntoinfty} a^n = 0 hbox{ if } |a| < 1
lim_{ntoinfty} n^{frac{1}{n}} = 1
lim_{ntoinfty} a^{frac{1}{n}} = 1 hbox{ if } a>0

Properties

Consider the following function: f(x)=xn if n-1<xn. Then the limit of the sequence of xn is just the limit of f(x) at infinity.

A function f, defined on a first-countable space, is continuous if and only if it is compatible with limits in that (f(xn)) converges to f(L) given that (xn) converges to L, i.e.

lim_{ntoinfty}x_n=L implies lim_{ntoinfty}f(x_n)=f(L)
Note that this equivalence does not hold in general for spaces which are not first-countable.

Compare the basic property (or definition):

f is continuous at x if and only if lim_{xto L}f(x)=f(L)

A subsequence of the sequence (xn) is a sequence of the form (xa(n)) where the a(n) are natural numbers with a(n) < a(n+1) for all n. Intuitively, a subsequence omits some elements of the original sequence. A sequence is convergent if and only if all of its subsequences converge towards the same limit.

Every convergent sequence in a metric space is a Cauchy sequence and hence bounded. A bounded monotonic sequence of real numbers is necessarily convergent: this is sometimes called the fundamental theorem of analysis. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete.

A sequence of real numbers is convergent if and only if its limit superior and limit inferior coincide and are both finite.

The algebraic operations are everywhere continuous (except for division around zero divisor); thus, given

lim_{n to infty}x_n = L_1 and lim_{n to infty}y_n = L_2

then

lim_{n to infty}(x_n+y_n) = L_1 + L_2

lim_{n to infty}(x_ny_n) = L_1L_2

and (if L2 and yn is non-zero)

lim_{n to infty}(x_n/y_n) = L_1/L_2

These rules are also valid for infinite limits using the rules

  • q + ∞ = ∞ for q ≠ -∞
  • q × ∞ = ∞ if q > 0
  • q × ∞ = -∞ if q < 0
  • q / ∞ = 0 if q ≠ ± ∞

(see extended real number line).

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 limits (letting o→0).

In the 18th century, mathematicians like 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 under which conditions a series converged to a limit.

The modern definition of a limit (for any ε there exists an index N so that ...) was given independently by Bernhard Bolzano (Der binomische Lehrsatz, Prag 1816, little noticed at the time) and by Cauchy in his Cours d'analyse (1821).

References

See also

External links

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