Sequences

Abstract

A sequence is simply a set whose elements are labelled by the positive integers (though a more formal definition is given in the next section). We write a sequence in the form s = (s₁,s₂, s₃,…) where the dots indicate that the list of terms continues indefinitely, so that any term, for instance s₄₉₁, is available for consideration if required. More explicitly we write s = (s₁, s₂, s₃,…, s_n,…) to indicate the n^th term, or simply s = (2n) ^∞₁ . But s_n alone (no parentheses!) is not the name of a sequence, it is the name of a number which is the n^th term of a sequence. For instance s = (2n - 1) ^∞₁ is the sequence of odd integers, s = (1,3,5,…) whose n^th term is s_n = 2n - 1.