Sequences and Series

Abstract

In the first chapter, we defined a sequence in X to be a mapping from N to X. Let us broaden this definition slightly, and allow the mapping to have a domain of the form $\left\{ {n \in {\rm Z}:m\underline < n\underline < p} \right\},or\left\{ {n \in {\rm Z}:n\underline > m} \right\}$, for some m ∈ Z (usually, but not always, m = 0 or m = 1). The most common notation is to write n→ x_n instead of n→ x(n). If the domain of the sequence is the finite set $\left\{ {m,m + 1, \ldots ,p} \right\}$, we write the sequence as $\left( {x_n } \right)_{n = m}^p$, and speak of a finite sequence (though we emphasize that the sequence should be distinguished from the set $\left. {\left\{ {x_n :m\underline < n\underline < p} \right\}} \right)$. If the domain of the sequence is a set of the form $\left\{ {m,m + 1,m + 2, \ldots } \right\} = \left\{ {n \in {\rm Z}:n\underline { > m} } \right\}$, we write it as $(x_n )_{n = m}^\infty$, and speak of an infinite sequence. Note that the corresponding set of values $\left\{ {x_n :n\underline > m} \right\}$ may be finite. When the domain of the sequence is understood from the context, or is not relevant to the discussion, we write simply (x_n). In this chapter, we shall be concerned with infinite sequences in R.