StatLect
Index > Mathematical tools

Sequences

Let A be a set of objects (e.g., real numbers, events, random variables). A sequence of elements of A is a function from the set of natural numbers $U{2115} $ to the set A, i.e., a correspondence that associates one and only one element of A to each natural number $nin U{2115} $. In other words, a sequence of elements of A is an ordered list of elements of A, where the ordering is provided by the natural numbers.

A sequence is usually indicated by enclosing a generic element of the sequence in curly brackets:[eq1]where a_n is the n-th element of the sequence. Alternative notations are[eq2]Thus, if [eq3] is a sequence, a_1 is its first element, a_2 is its second element, a_n is its n-th element, and so on.

Example Define a sequence [eq3] by characterizing its n-th element a_n as follows:[eq5][eq3] is a sequence of rational numbers. The elements of the sequence are $a_{1}=1$, $frac{1}{2}$, $frac{1}{3}$, $frac{1}{4}$ and so on.

Example Define a sequence [eq3] by characterizing its n-th element a_n as follows:[eq8][eq3] is a sequence of 0 and 1. The elements of the sequence are $a_{1}=0$, $a_{2}=1$, $a_{3}=0$, $a_{4}=1$ and so on.

Example Define a sequence [eq3] by characterizing its n-th element a_n as follows:[eq11][eq3] is a sequence of closed subintervals of the interval $left[ 0,1
ight] $. The elements of the sequence are [eq13], [eq14], [eq15], [eq16] and so on.

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Countable and uncountable sets

Let A be a set of objects. A is a countable set if all its elements can be arranged into a sequence, i.e., if there exists a sequence [eq17] such that[eq18]In other words, A is a countable set if there exists at least one sequence [eq3] such that every element of A belongs to the sequence. A is an uncountable set if such a sequence does not exist. The most important example of an uncountable set is the set of real numbers R.

Limit of a sequence

The concept of limit of a sequence is discussed in the lecture entitled Limit of a sequence.

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