Limit of a sequence

Limit of a sequence is one of the oldest concept in mathematical analysis. It is the essential tool in calculating pi and trigonometric functions.

Table of contents

1 History
2 Formal definition
3 Examples
4 Properties
5 See Also

History

See mathematical analysis.

Formal definition

Suppose x₁, x₂, ... is a sequence of elementss in a metric space (M,d). We say that the real number L is the limit of this sequence and we write

if and only if

for every ε>0 there exists a natural number n₀ (which will depend on ε) such that for all n>n₀ we have d(x_n,L) < ε.

Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. One can show that a convergent sequence has only one limit.

For sequence of real or complex numbers, the metric (distance) between x_n and L is the absolute value |x_n - L|.

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 a^1/n has limit 1.

Properties

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 infinitely.

A function f : R -> R is continuous if and only if it is compatible with limits in the following sense:

if (x_n) is any convergent sequence in R with limit L, then the sequence (f(x_n)) converges with limit f(L).

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 is a Cauchy sequence and hence bounded. If (x_n) is a bounded sequence of real numbers which is increasing (i.e. x_n ≤ x_n+1 for all n), then it 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

and

then

and

(the latter provided that f₂(x) is non-zero in a neighborhood of p and L₂ is non-zero as well).

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).