Mathematical Analysis pp 28-54 | Cite as

# 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.

## Keywords

Cauchy Sequence Continue Fraction Infinite Series Infinite Sequence Convergent Sequence## Preview

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