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 |
2 Formal definition 3 Examples 4 Properties 5 See Also |
History
See mathematical analysis.Formal definition
Suppose x_{1}, x_{2}, ... 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
- for every ε>0 there exists a natural number n_{0} (which will depend on ε) such that for all n>n_{0} we have d(x_{n},L) < ε.
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 a^{n} 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).
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
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 ≠ ± ∞