Wikipedia / Limit of a sequence
Wikipedia / Limit of a sequence
As the positive integer."
In mathematics, the limit of a sequence is the value that the terms of a sequence). If such a limit exists and is finite, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.
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 in his Quadrature of the Parabola, computing the area enclosed by a parabola and a straight line.
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."
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, which he then linearizes by taking the limit.
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.
Real numbers
In the real numbers is the limit of the sequence, and not to any other number.
Examples
Examples of limit of a sequence in real numbers are the following:
.
.
is odd is irrelevant.)
. The decimal representation is the limit
(the limit of which is the number e) and the arithmetic–geometric mean. The squeeze theorem is often useful in the establishment of such limits.
Definition
the limit of the sequence, which is written
, or
,
if the following condition holds:
For each real number, there exists a natural number.
is said to converge to or tend to.
Symbolically, this is:
.
, then it is convergent is divergent. A sequence that has zero as its limit is sometimes called a null sequence.
Illustration
Properties
Some other important properties of limits of real sequences include the following:
When it exists, the limit of a sequence is unique.
Limits of sequences behave well with respect to the usual arithmetic operations exists, then
For any continuous function exists too. In fact, any real-valued function is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity).
.
(Monotone convergence theorem is bounded and monotonic, then it is convergent.
A sequence is convergent if and only if every subsequence is convergent.
If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.
).
Infinite limits
is said to tend to infinity, written
, or
,
if the following holds:
.
Symbolically, this is:
.
Similarly, we say a sequence tends to minus infinity, written
, or
,
if the following holds:
.
Symbolically, this is:
.
provides one such example.
Metric spaces
Definition
of the metric space is the limit of the sequence if:
For each real number, there is a natural number.
Symbolically, this is:
.
.
Properties
of both points.
For any continuous function f. In fact, a function f is continuous if and only if it preserves the limits of sequences.
Cauchy sequences
A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is the Cauchy criterion for convergence of sequences: a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains true in other complete metric spaces.
Topological spaces
Definition
is a or of the sequence if:
For every neighbourhood.
.
is a special case of a limit of a function: the domain, with the induced topology of the affinely extended real number system, the range, which in this space is a limit point.
Properties
In a Hausdorff space are topologically indistinguishable and vice versa.
Hyperreal numbers
The definition of the limit using the hyperreal numbers tends to L if for every infinite hypernatural is infinitesimal). Equivalently, L is the standard part:
.
Thus, the limit can be defined by the formula
.
where the limit exists if and only if the righthand side is independent of the choice of an infinite .
Sequence of more than one index
when both n and m becomes very large.
Example
.
.
.
Definition
the double limit of the sequence, written
, or
,
if the following condition holds:
For each real number, there exists a natural number.
is said to converge to or tend to.
Symbolically, this is:
.
The double limit is different from taking limit in n first, and then in m. The latter is known as iterated limit. Given that both the double limit and the iterated limit exists, they have the same value. However, it is possible that one of them exist but the other does not.
Infinite limits
is said to tend to infinity, written
, or
,
if the following holds:
.
Symbolically, this is:
.
tends to minus infinity, written
, or
,
if the following holds:
.
Symbolically, this is:
.
provides one such example.
Pointwise limits and uniform limits
. In fact, there are two possible meanings when taking this limit. The first one is called pointwise limit, denoted
, or
,
which means:
For each real number and each fixed natural number.
Symbolically, this is:
.
The second one is called uniform limit, denoted
,
,
, or
,
which means:
For each real number such that, for every natural number.
Symbolically, this is:
.
is uniformly applicable. 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:
pointwise.
Iterated limit
. Symbolically,
.
This limit is known as iterated limit of the double sequence. The order of taking limits may affect the result, i.e.,
in general.
A sufficient condition of equality is given by the Moore-Osgood theorem.
See also
Notes
Proofs
References
Courant, Richard (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow.
Frank Morley and James Harkness A treatise on the theory of functions (New York: Macmillan, 1893)
External links