Limit of a sequence

[1] If such a limit exists and is finite, the sequence is called convergent.

[3] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.

[1] Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.

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.

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment.

"[4] Pietro Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions in Geometriae speciosae elementa (1659).

He used the term quasi-infinite for unbounded and quasi-null for vanishing.

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

In the 18th century, mathematicians such as 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 study of hypergeometric series (1813) for the first time rigorously investigated the conditions under which a series converged to a limit.

so that ...) was given by Bernard Bolzano (Der binomische Lehrsatz, Prague 1816, which was little noticed at the time), and by Karl Weierstrass in the 1870s.

, which is written if the following condition holds: In other words, for every measure of closeness

, the sequence's terms are eventually that close to the limit.

Some other important properties of limits of real sequences include the following: These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition.

if: Symbolically, this is: This coincides with the definition given for real numbers when

A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded.

This remains true in other complete metric spaces.

is a special case of a limit of a function: the domain is

, with the induced topology of the affinely extended real number system, the range is

In a Hausdorff space, limits of sequences are unique whenever they exist.

, written if the following condition holds: In other words, for every measure of closeness

, the sequence's terms are eventually that close to the limit.

is said to tend to infinity, written if the following holds: Symbolically, this is: Similarly, a sequence

The second one is called uniform limit, denoted which means: Symbolically, this is: In this definition, the choice of

Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of pointwise limit: When such a limit exists, we say the sequence

The order of taking limits may affect the result, i.e., A sufficient condition of equality is given by the Moore-Osgood theorem, which requires the limit

diagram of a hexagon and pentagon circumscribed outside a circle
The sequence given by the perimeters of regular n -sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. . The corresponding sequence for inscribed polygons has the same limit.
The plot of a convergent sequence { a n } is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases.
The plot of a Cauchy sequence ( x n ), shown in blue, as versus n . Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence converges to some limit.