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 $\sum_{n = 1}^{\infty} {(−1)}^{n + 1} b_{n}$ or $\sum_{n = 1}^{\infty} {(−1)}^{n} b_{n},$ where $b_{n} \geq 0,$ is called an alternating series

alternating series test: for an alternating series of either form, if $b_{n + 1} \leq b_{n}$ for all integers $n \geq 1$ 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 ${a_{n}}$ is bounded above if there exists a constant $M$ such that $a_{n} \leq M$ for all positive integers $n$

bounded below: a sequence ${a_{n}}$ is bounded below if there exists a constant $M$ such that $M \leq a_{n}$ for all positive integers $n$

bounded sequence: a sequence ${a_{n}}$ is bounded if there exists a constant $M$ such that $| a_{n} | \leq M$ for all positive integers $n$

comparison test: if $0 \leq a_{n} \leq b_{n}$ for all $n \geq N$ and $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges; if $a_{n} \geq b_{n} \geq 0$ for all $n \geq N$ 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 ${a_{n}}$ 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 $lim_{n \to \infty} a_{n} \neq 0,$ then the series $\sum_{n = 1}^{\infty} a_{n}$ diverges

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

geometric sequence: a sequence ${a_{n}}$ in which the ratio $a_{n + 1} / a_{n}$ 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

$\sum_{n = 1}^{\infty} a r^{n - 1} = a + a r + a r^{2} + a r^{3} + ⋯$

harmonic series: the harmonic series takes the form

$\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ⋯$

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

$a_{1} + a_{2} + a_{3} + ⋯ = \sum_{n = 1}^{\infty} a_{n}$

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 $f (n) = a_{n}$ for all positive integers $n,$ then

$\sum_{n = 1}^{\infty} a_{n} and \int_{1}^{\infty} f (x) d x$

either both converge or both diverge

limit comparison test: suppose $a_{n}, b_{n} \geq 0$ for all $n \geq 1 .$ If $lim_{n \to \infty} a_{n} / b_{n} \to L \neq 0,$ then $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ both converge or both diverge; if $lim_{n \to \infty} a_{n} / b_{n} \to 0$ and $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges. If $lim_{n \to \infty} a_{n} / b_{n} \to \infty,$ 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

partial sum: the $k th$ partial sum of the infinite series $\sum_{n = 1}^{\infty} a_{n}$ is the finite sum

$S_{k} = \sum_{n = 1}^{k} a_{n} = a_{1} + a_{2} + a_{3} + ⋯ + a_{k}$

ratio test: for a series $\sum_{n = 1}^{\infty} a_{n}$ with nonzero terms, let $ρ = lim_{n \to \infty} | a_{n + 1} / a_{n} |;$ if $0 \leq ρ < 1,$ the series converges absolutely; if $ρ > 1,$ the series diverges; if $ρ = 1,$ 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 $f (n) = a_{n}$ for all positive integers $n,$ the remainder $R_{N} = \sum_{n = 1}^{\infty} a_{n} - \sum_{n = 1}^{N} a_{n}$ satisfies the following estimate:

$\int_{N + 1}^{\infty} f (x) d x < R_{N} < \int_{N}^{\infty} f (x) d x$

root test: for a series $\sum_{n = 1}^{\infty} a_{n},$ let $ρ = lim_{n \to \infty} \sqrt[n]{| a_{n} |};$ if $0 \leq ρ < 1,$ the series converges absolutely; if $ρ > 1,$ the series diverges; if $ρ = 1,$ the test is inconclusive

sequence: an ordered list of numbers of the form $a_{1}, a_{2}, a_{3},…$ 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 ${a_{n}}$ is called the $n th$ term of the sequence