Summary of Sequences

To determine the convergence of a sequence given by an explicit formula ${a}_{n}=f\left(n\right)$ , we use the properties of limits for functions.
If $\left\{{a}_{n}\right\}$ and $\left\{{b}_{n}\right\}$ are convergent sequences that converge to $A$ and $B$ , respectively, and $c$ is any real number, then the sequence $\left\{c{a}_{n}\right\}$ converges to $c\cdot A$ , the sequences $\left\{{a}_{n}\pm {b}_{n}\right\}$ converge to $A\pm B$ , the sequence $\left\{{a}_{n}\cdot {b}_{n}\right\}$ converges to $A\cdot B$ , and the sequence $\left\{\frac{{a}_{n}}{{b}_{n}}\right\}$ converges to $\frac{A}{B}$ , provided $B\ne 0$ .
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 $\left\{{r}^{n}\right\}$ converges if and only if $|r|<1$ or $r=1$ .

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 $\left\{{a}_{n}\right\}$ is bounded above if there exists a constant $M$ such that ${a}_{n}\le M$ for all positive integers $n$

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

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

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

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

geometric sequence: a sequence $\left\{{a}_{n}\right\}$ in which the ratio $\frac{{a}_{n+1}}{{a}_{n}}$ is the same for all positive integers $n$ 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 $L$ to which a sequence converges is called the limit of the sequence

recurrence relation: a recurrence relation is a relationship in which a term ${a}_{n}$ in a sequence is defined in terms of earlier terms in the sequence

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

term: the number ${a}_{n}$ in the sequence $\left\{{a}_{n}\right\}$ is called the $n\text{th}$ term of the sequence