Sequences

Abstract

A sequence is a function whose domain is a set of the form {n ∈ ℤ : n ⩾ m}; m is usually 1 or 0. Thus a sequence is a function that has a specified value for each integer n ⩾ m. It is customary to denote a sequence by a letter such as s and to denote its value at n as s _n rather than s(n). It is often convenient to write the sequence as $({S_n})_{n = m}^\infty or({S_m},{S_{m + 1}},{S_{m + 2}},...)$. If m = 1 we may write (s _n ) _n ∈ ℕ or of course (s ₁,s ₂,s ₃,...). Sometimes we will write (s _n ) when the domain is understood or when the results under discussion do not depend on the specific value of m. In this chapter we will be interested in sequences whose range values are real numbers, i.e., each s _n represents a real number.