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Section 11.1 Sequences
counting set
the set
\begin{equation*}
\natnumlt{m} = \setdef{n \in \N}{n \lt m} = \{ 0,\, 1,\, \dotsc,\, m-1\}
\end{equation*}
The set
\(\natnumlt{m}\) 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.
finite sequence (from a set \(A\) )
a function \(\natnumlt{m} \to A\)
infinite sequence (from a set \(A\) )
a function \(\N \to A\)
term in a sequence
one of the image elements of the function defining the sequence
\(a_k\) the \(\nth[k]\) term in a sequence, so that if \(\funcdef{f}{\natnumlt{m}}{A}\) or \(\funcdef{f}{\N}{A}\) is a sequence then \(a_k = f(k)\)
\(\{a_k\}\) the collection of all terms in a sequence
\(\{a_k\}_0^m\) the collection of the terms in a sequence up to (and including) the \(\nth[m]\) term (if the sequence is finite, this could represent all terms in the sequence for the appropriate \(m\) value)
\(\{a_k\}_0^\infty\) the collection of all terms in a sequence, where we are explicit that it is an infinite sequence
Example 11.1.2 . Sequence of squares.
The sequence \(\{k^2\}\) has terms \(0, \, 1, \, 4, \, 9, \, 16, \, 25, \, \dotsc, \, k^2, \, \dotsc\text{.}\)
Example 11.1.3 . Sequence of definite integral values.
The sequence
\begin{equation*}
\left\{ (-1)^k\int_1^{k+1} \frac{dx}{x} \right\}
\end{equation*}
has terms \(0, \, -\ln 2, \, \ln 3, \, -\ln 4, \, \dotsc, \, (-1)^k\,\ln (k+1), \, \dotsc \text{.}\)
