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Sequences and Series

  • Andrew Browder
Part of the Undergraduate Texts in Mathematics book series (UTM)

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 nxn instead of nx(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 (xn). In this chapter, we shall be concerned with infinite sequences in R.

Keywords

Cauchy Sequence Continue Fraction Infinite Series Infinite Sequence Convergent Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Andrew Browder
    • 1
  1. 1.Mathematics DepartmentBrown UniversityProvidenceUSA

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