Limit of a sequence
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Limit of a sequence is one of the oldest concepts in mathematical analysis. It is the essential tool in calculating pi and trigonometric functions.
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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 Théorie 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: [1] (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 a limit of this sequence and write
- <math> \lim_{n \to \infty} x_n = L <math>
if and only if
- for every neighborhood S of L there is an N such that xn∈S for all n>N.
If a sequence has a limit, we call it convergent, otherwise divergent.
Comments
The definition means that eventually all elements of the sequence get as close as we want to the limit. (This does not imply that there is a limit whenever the elements become as close as we want to all of the following elements, see Cauchy sequence).
Also, a sequence may have several different limits, but a convergent sequence has a unique limit if T is a Hausdorff space, as 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 a1/n has limit 1.
Properties
Consider the following function: f(x)=xn if n-1<x≤n. Then the limit of the sequence of xn 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 (xn) is any convergent sequence in R with limit L, then the sequence (f(xn)) converges with limit 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. 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
- <math>
\lim_{n \to \infty}x_n = L_1 <math> and
- <math>
\lim_{n \to \infty}y_n = L_2 <math>
then
- <math>
\lim_{n \to \infty}(x_n+y_n) = L_1 + L_2 <math>
- <math>
\lim_{n \to \infty}(x_ny_n) = L_1L_2 <math>
and (if L2 is non-zero)
- <math>
\lim_{n \to \infty}(x_n/y_n) = L_1/L_2 <math>
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).