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.

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. 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 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].

Formal definition

Suppose x1, x2, ... is a sequence of elements in a topological space T. We say that LT is a limit of this sequence and write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} x_n = L }

if and only if

for every neighborhood S of L there is an N such that xnS 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.

  • for normed spaces : if ||.|| is the norm,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} x_n = L } if and only if for every e>0,there is an N natural number so that for every n>N, we have ||xn-L||<e
  • for metric spaces : if d is the distance,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} x_n = L } if and only if for every e>0,there is an N natural number so that for every n>N, we have d(xn,L)<e

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<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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty}x_n=L} implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 iff Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 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 also known at 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty}x_n = L_1 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty}y_n = L_2 }

then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty}(x_n+y_n) = L_1 + L_2 }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty}(x_ny_n) = L_1L_2 }

and (if L2 is non-zero)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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).

See also

External References

Examples of Sequences

lt:Sekos riba pl:Granica ciągu ru:Предел последовательности