Undergraduate Mathematics/Limit of a sequence
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n | n sin(1/n) |
---|---|
1 | 0.841471 |
2 | 0.958851 |
... | |
10 | 0.998334 |
... | |
100 | 0.999983 |
The limit of a sequence is, intuitively, the unique number or point L (if it exists) such that the terms of the sequence become arbitrarily close to L for "large" values of n. If the limit exists, then we say that the sequence is convergent and that it converges to L.
Convergence of sequences is a fundamental notion in mathematical analysis, which has been studied since ancient times.
Formal definition[edit]
- For a sequence of real numbers
- A real number L is said to be the limit of the sequence x_{n}, written
- if and only if for every real number there exists a natural number such that for every natural number we have .
- As a generalization of this, for a sequence of points in a topological space T:
- An element L ∈ T is said to be a limit of this sequence if and only if for every neighborhood S of L there is a natural number N such that x_{n} ∈ S for all n ≥ N. In this generality a sequence may admit more than one limit, but if T is a Hausdorff space, then each sequence has at most one limit. When a unique limit L exists, it may be written
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 (see also oscillation).
A null sequence is a sequence that converges to 0.
Hyperreal definition[edit]
A sequence x_{n} tends to L if for every infinite hypernatural H, the term x_{H} is infinitely close to L, i.e., the difference x_{H} - L is infinitesimal. Equivalently, L is the standard part of x_{H}
- .
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 H.
Comments[edit]
The definition means that eventually all elements of the sequence get as close as we want to the limit. (The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence).
A sequence of real numbers may tend to or , compare infinite limits. Even though this can be written in the form
- and
such a sequence is called divergent, unless we explicitly consider it a sequence in the affinely extended real number system or (in the first case only) the real projective line. In the latter cases the sequence has a limit (in the space itself), so could be called convergent, but when using this term here, care should be taken that this does not cause confusion.
The limit of a sequence of points in a topological space T is a special case of the limit of a function: the domain is in the space with the induced topology of the affinely extended real number system, the range is T, and the function argument n tends to +∞, which in this space is a limit point of .
Examples[edit]
- The sequence 1, -1, 1, -1, 1, ... is oscillatory.
- The series with partial sums 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 a^{n} has limit 0. If 0 < a, then the sequence a^{1/n} has limit 1.
Also:
Properties[edit]
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, defined on a first-countable space, is continuous if and only if it is compatible with limits in that (f(x_{n})) converges to f(L) given that (x_{n}) converges to L, i.e.
- implies
Note that this equivalence does not hold in general for spaces which are not first-countable.
Compare the basic property (or definition):
- f is continuous at x if and only if
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: this is sometimes called the fundamental theorem of analysis. 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 superior and limit inferior coincide and are both finite.
The algebraic operations are continuous everywhere (except for division around zero divisor); thus, given
- and
then
and (if L_{2} and y_{n} are non-zero)
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).
History[edit]
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.
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, mathematicians 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 analytiques (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, Prague 1816, little noticed at the time) and by Cauchy in his Cours d'analyse (1821).
References[edit]
- Frank Morley and James Harkness A treatise on the theory of functions (New York: Macmillan, 1893)