### Key Terms

- absolute convergence
- if the series $\sum}_{n=1}^{\infty}|{a}_{n}|$ converges, the series $\sum}_{n=1}^{\infty}{a}_{n$ is said to converge absolutely

- alternating series
- a series of the form $\sum}_{n=1}^{\infty}{\left(\mathrm{-1}\right)}^{n+1}{b}_{n$ or $\sum}_{n=1}^{\infty}{\left(\mathrm{-1}\right)}^{n}{b}_{n},$ where ${b}_{n}\ge 0,$ is called an alternating series

- alternating series test
- for an alternating series of either form, if ${b}_{n+1}\le {b}_{n}$ for all integers $n\ge 1$ and ${b}_{n}\to 0,$ then an alternating series converges

- 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 $\left|{a}_{n}\right|\le M$ for all positive integers $n$

- comparison test
- if $0\le {a}_{n}\le {b}_{n}$ for all $n\ge N$ and $\sum}_{n=1}^{\infty}{b}_{n$ converges, then $\sum}_{n=1}^{\infty}{a}_{n$ converges; if ${a}_{n}\ge {b}_{n}\ge 0$ for all $n\ge N$ and $\sum}_{n=1}^{\infty}{b}_{n$ diverges, then $\sum}_{n=1}^{\infty}{a}_{n$ diverges

- conditional convergence
- if the series $\sum}_{n=1}^{\infty}{a}_{n$ converges, but the series $\sum}_{n=1}^{\infty}|{a}_{n}|$ diverges, the series $\sum}_{n=1}^{\infty}{a}_{n$ is said to converge conditionally

- convergence of a series
- a series converges if the sequence of partial sums for that series converges

- 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

- divergence of a series
- a series diverges if the sequence of partial sums for that series diverges

- divergence test
- if $\underset{n\to \infty}{\text{lim}}{a}_{n}\ne 0,$ then the series $\sum}_{n=1}^{\infty}{a}_{n$ diverges

- divergent sequence
- a sequence that is not convergent is divergent

- 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 ${a}_{n+1}\text{/}{a}_{n}$ is the same for all positive integers $n$ is called a geometric sequence

- geometric series
- a geometric series is a series that can be written in the form

$$\sum}_{n=1}^{\infty}a{r}^{n-1}=a+ar+a{r}^{2}+a{r}^{3}+\text{\cdots$$

- harmonic series
- the harmonic series takes the form

$$\sum}_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\text{\cdots$$

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

- infinite series
- an infinite series is an expression of the form

$${a}_{1}+{a}_{2}+{a}_{3}+\text{\cdots}={\displaystyle \sum}_{n=1}^{\infty}{a}_{n}$$

- integral test
- for a series $\sum}_{n=1}^{\infty}{a}_{n$ with positive terms ${a}_{n},$ if there exists a continuous, decreasing function $f$ such that $f\left(n\right)={a}_{n}$ for all positive integers $n,$ then

$$\sum}_{n=1}^{\infty}{a}_{n}\text{and}{\displaystyle {\int}_{1}^{\infty}f\left(x\right)dx$$

either both converge or both diverge

- limit comparison test
- suppose ${a}_{n},\phantom{\rule{0.2em}{0ex}}{b}_{n}\ge 0$ for all $n\ge 1.$ If $\underset{n\to \infty}{\text{lim}}{a}_{n}\text{/}{b}_{n}\to L\ne 0,$ then $\sum}_{n=1}^{\infty}{a}_{n$ and $\sum}_{n=1}^{\infty}{b}_{n$ both converge or both diverge; if $\underset{n\to \infty}{\text{lim}}{a}_{n}\text{/}{b}_{n}\to 0$ and $\sum}_{n=1}^{\infty}{b}_{n$ converges, then $\sum}_{n=1}^{\infty}{a}_{n$ converges. If $\underset{n\to \infty}{\text{lim}}{a}_{n}\text{/}{b}_{n}\to \infty ,$ and $\sum}_{n=1}^{\infty}{b}_{n$ diverges, then $\sum}_{n=1}^{\infty}{a}_{n$ diverges

- limit of a sequence
- the real number $L$ to which a sequence converges is called the limit of the sequence

- monotone sequence
- an increasing or decreasing sequence

*p*-series- a series of the form $\sum}_{n=1}^{\infty}1\text{/}{n}^{p$

- partial sum
- the $k\text{th}$ partial sum of the infinite series $\sum}_{n=1}^{\infty}{a}_{n$ is the finite sum

$${S}_{k}={\displaystyle \sum}_{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\text{\cdots}+{a}_{k}$$

- ratio test
- for a series $\sum}_{n=1}^{\infty}{a}_{n$ with nonzero terms, let $\rho =\underset{n\to \infty}{\text{lim}}\left|{a}_{n+1}\text{/}{a}_{n}\right|;$ if $0\le \rho <1,$ the series converges absolutely; if $\rho >1,$ the series diverges; if $\rho =1,$ the test is inconclusive

- 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

- remainder estimate
- for a series $\sum}_{n=1}^{\infty}{a}_{n$ with positive terms ${a}_{n}$ and a continuous, decreasing function $f$ such that $f\left(n\right)={a}_{n}$ for all positive integers $n,$ the remainder ${R}_{N}={\displaystyle \sum}_{n=1}^{\infty}{a}_{n}-{\displaystyle \sum}_{n=1}^{N}{a}_{n}$ satisfies the following estimate:

$${\int}_{N+1}^{\infty}f\left(x\right)dx<{R}_{N}<}{\displaystyle {\int}_{N}^{\infty}f\left(x\right)dx$$

- root test
- for a series $\sum}_{n=1}^{\infty}{a}_{n},$ let $\rho =\underset{n\to \infty}{\text{lim}}\sqrt[n]{\left|{a}_{n}\right|};$ if $0\le \rho <1,$ the series converges absolutely; if $\rho >1,$ the series diverges; if $\rho =1,$ the test is inconclusive

- sequence
- an ordered list of numbers of the form ${a}_{1},{a}_{2},{a}_{3}\text{,\u2026}$ is a sequence

- telescoping series
- a telescoping series is one in which most of the terms cancel in each of the partial sums

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

- unbounded sequence
- a sequence that is not bounded is called unbounded