11.1: Sequences

Elementary Foundations: An Introduction to Topics in Discrete Mathematics · LibreTexts

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    Definition: counting set

    the set

    \begin{equation*} \mathbb{N}_{<m} = \{n \in \mathbb{N} \vert n \lt m \} = \{ 0,\, 1,\, \ldots ,\, m-1\} \end{equation*}

    The set \(\mathbb{N}_{<m}\) has exactly \(m\) elements in it. In Chapter 12 we will use these counting sets to, well, count the elements in other sets. For now, we will use them to index the objects in an ordered list.

    Definition: Finite sequence(from a set \(A\))

    a function \(\mathbb{N}_{<m} \to A\)

    Definition: infinite sequence (from a set \(A\))

    a function \(\mathbb{N} \to A\)

    Definition: term in a sequence

    one of the image elements of the function defining the sequence

    Definition: \(a_k\)

    the \(k^{th}\) term in a sequence, so that if \(f: \mathbb{N}_{<m} \rightarrow A\) or \(f: \mathbb{N} \rightarrow A\) is a sequence then \(a_k = f(k)\)

    Definition: \(\{a_k\}\)

    the collection of all terms in a sequence

    Definition: \(\{a_k\}_0^m\)

    the collection of the terms in a sequence up to (and including) the \(m^{th}\) term (if the sequence is finite, this could represent all terms in the sequence for the appropriate \(m\) value)

    Definition: \(\{a_k\}_0^\infty\)

    the collection of all terms in a sequence, where we are explicit that it is an infinite sequence

    Remark

    • Of course, we do not restrict ourselves to the letter \(a\) to represent the terms of a sequence. We might write \(b_k\text{,}\) or \(s_k\text{,}\) etc..
    • While we use set-like notation \(\{\}\) to represent the collection of all terms in a sequence, this collection is not a set, since order and repetition matter.

    Example \(\PageIndex{1}\): Sequence of squares.

    The sequence \(\{k^2\}\) has terms \(0, \, 1, \, 4, \, 9, \, 16, \, 25, \, \ldots , \, k^2, \, \ldots \text{.}\)

    Example \(\PageIndex{2}\): Sequence of definite integral values.

    The sequence

    \begin{equation*} \left\{ (-1)^k\int_1^{k+1} \dfrac{dx}{x} \right\} \end{equation*}
    has terms \(0, \, -\ln 2, \, \ln 3, \, -\ln 4, \, \ldots , \, (-1)^k\,\ln (k+1), \, \ldots \text{.}\)

    This page titled 11.1: Sequences was last modified on Tue, 01 Sep 2026 03:35:25 GMT and is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.