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Sequences

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Elementary Analysis

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Abstract

Asequence is a function whose domain is a set of the form\(\{n \in\mathbb{Z} : n \geq m\}\);m is usually 1 or 0. Thus a sequence is a function that has a specified value for each integernm. It is customary to denote a sequence by a letter such ass and to denote its value atn ass n rather thans(n). It is often convenient to write the sequence as (s n ) n = m or\((s_{m},s_{m+1},s_{m+2},\ldots )\)

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Notes

  1. 1.

    This exercise is referred to in several places.

  2. 2.

    This exercise is referred to in several places.

  3. 3.

    This exercise is referred to in several places.

  4. 4.

    In the First Edition of this book, increasing and decreasing sequences were referred to as “nondecreasing” and “nonincreasing” sequences, respectively.

  5. 5.

    In the first edition of this book, we did create similar proofs instead.

  6. 6.

    This will be proved easily here, but is also a consequence of the more general Theorem 11.4.

  7. 7.

    Recursive definitions of sequences, which first appear in Exercises 9.4–9.6, can be viewed as simple examples of definitions by induction.

  8. 8.

    This exercise is referred to in several places.

  9. 9.

    As noted in [35], the proofs of this corollary and the Alternating Series Theorem 15.3 use the completeness of \(\mathbb{R}\).

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Ross, K.A. (2013). Sequences. In: Elementary Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6271-2_2

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