limit of a sequence

Limit of a sequence is one of the oldest concepts in mathematical analysis. It is the essential tool in calculating pi and trigonometric functions.

History

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involved 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. Archmides 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 Thorie des fonctions analytique (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: http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html.

Formal definition

Suppose x₁, x₂, ... is a sequence of elements in a topological space T. We say that L∈T is the 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 an N such that x_n∈S for all n>N.

Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. In a Hausdorff space, a convergent sequence has a unique limit.

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.

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 : R -> R is continuous if and only if it is compatible with limits in the following sense:

if (x_n) is any convergent sequence in R with limit L, then the sequence (f(x_n)) converges with limit 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. 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. Taking the limit of sequences is compatible with the algebraic operations: If

\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 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).