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 balls, only a finite number of points in the sequence being outside the ball.

Formal definition

For a sequence of points $\{x_n|n\in \mathbb{N}\}\;$ in a metric space M with distance function d

(such as a sequence of rational numbers, real numbers, complex numbers, points in a normed space, etc.):

L\in M\;

we say L is the limit of the sequence and write

\lim_{n \to \infty} x_n = L \;

if and only if for every real number

\epsilon>0\;

, there is a natural number N so that for every

n>N\;

, we have

d(x_n,L)<\epsilon\;

For a sequence of points $\{x_n|n\in \mathbb{N}\}\;$ in a topological space T:

L\in T\;

we say L is a limit of this sequence and write

\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_n\in 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.

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).

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...).

Examples

The sequence 1/1, 1/2, 1/3, 1/4, ... of real numbers converges with limit 0.
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 a^1/n has limit 1.

Also:

$\lim_{n\to\infty} \frac{1}{n^p} = 0 | p \neq 0$
$\lim_{n\to\infty} a^n = 0 | |a| < 1$
$\lim_{n\to\infty} n^{\frac{1}{n}} = 1$
$\lim_{n\to\infty} a^{\frac{1}{n}} = 1 | a>0$

Properties

Consider the following function: f(x)=x_n if n-1<x≤n. Then the limit of the sequence of x_n 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(x_n)) converges to f(L) given that (x_n) converges to L, i.e.

\lim_{n\to\infty}x_n=L

implies

\lim_{n\to\infty}f(x_n)=f(L)

Note that this equivalence does not hold in general for spaces which are not first-countable. For functions in the reals, this is often simplified to

f is continuous at x if and only if

\lim_{x\to L}f(x)=f(L)

A subsequence of the sequence (x_n) is a sequence of the form (x_a(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 known as the monotone convergence theorem. 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 inferior and limit superior 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 y_n 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, virtuosi 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).

See also mathematical analysis; external link: ^*.

External links

Examples of Sequences

Mathematical analysis | Sequences