Summary of Sequences

To determine the convergence of a sequence given by an explicit formula [latex]{a}_{n}=f\left(n\right)[/latex], we use the properties of limits for functions.
If [latex]\left\{{a}_{n}\right\}[/latex] and [latex]\left\{{b}_{n}\right\}[/latex] are convergent sequences that converge to [latex]A[/latex] and [latex]B[/latex], respectively, and [latex]c[/latex] is any real number, then the sequence [latex]\left\{c{a}_{n}\right\}[/latex] converges to [latex]c\cdot A[/latex], the sequences [latex]\left\{{a}_{n}\pm {b}_{n}\right\}[/latex] converge to [latex]A\pm B[/latex], the sequence [latex]\left\{{a}_{n}\cdot {b}_{n}\right\}[/latex] converges to [latex]A\cdot B[/latex], and the sequence [latex]\left\{\frac{{a}_{n}}{{b}_{n}}\right\}[/latex] converges to [latex]\frac{A}{B}[/latex], provided [latex]B\ne 0[/latex].
If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone.
If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded.
The geometric sequence [latex]\left\{{r}^{n}\right\}[/latex] converges if and only if [latex]|r|<1[/latex] or [latex]r=1[/latex].

Glossary

arithmetic sequence: a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence

bounded above: a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded above if there exists a constant [latex]M[/latex] such that [latex]{a}_{n}\le M[/latex] for all positive integers [latex]n[/latex]

bounded below: a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded below if there exists a constant [latex]M[/latex] such that [latex]M\le {a}_{n}[/latex] for all positive integers [latex]n[/latex]

bounded sequence: a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded if there exists a constant [latex]M[/latex] such that [latex]|{a}_{n}|\le M[/latex] for all positive integers [latex]n[/latex]

convergent sequence: a convergent sequence is a sequence [latex]\left\{{a}_{n}\right\}[/latex] for which there exists a real number [latex]L[/latex] such that [latex]{a}_{n}[/latex] is arbitrarily close to [latex]L[/latex] as long as [latex]n[/latex] is sufficiently large

explicit formula: a sequence may be defined by an explicit formula such that [latex]{a}_{n}=f\left(n\right)[/latex]

geometric sequence: a sequence [latex]\left\{{a}_{n}\right\}[/latex] in which the ratio [latex]\frac{{a}_{n+1}}{{a}_{n}}[/latex] is the same for all positive integers [latex]n[/latex] is called a geometric sequence

index variable: the subscript used to define the terms in a sequence is called the index

limit of a sequence: the real number [latex]L[/latex] to which a sequence converges is called the limit of the sequence

recurrence relation: a recurrence relation is a relationship in which a term [latex]{a}_{n}[/latex] in a sequence is defined in terms of earlier terms in the sequence

sequence: an ordered list of numbers of the form [latex]{a}_{1},{a}_{2},{a}_{3}\text{,}\ldots[/latex] is a sequence

term: the number [latex]{a}_{n}[/latex] in the sequence [latex]\left\{{a}_{n}\right\}[/latex] is called the [latex]n\text{th}[/latex] term of the sequence