Glossary

Alternating Sequence

Criticality: 2

A sequence in which the terms alternate in sign, often identified by a $(-1)^n$ or $(-1)^{n+1}$ factor in its formula.

Example:

The sequence ${(-1)^n/n}^\infty_1$ produces terms like -1, 1/2, -1/3, 1/4, ..., demonstrating an alternating sequence.

Bounded (Sequence)

Criticality: 2

A sequence that is both bounded above and bounded below.

Example:

The sequence ${(-1)^n/n}$ is a bounded sequence because its terms are always between -1 and 1 (inclusive).

Bounded Above

Criticality: 1

A sequence is bounded above if there exists some real number M such that all terms of the sequence are less than or equal to M.

Example:

The sequence ${1/n}$ is bounded above by 1, as no term in the sequence will ever exceed 1.

Bounded Below

Criticality: 1

A sequence is bounded below if there exists some real number m such that all terms of the sequence are greater than or equal to m.

Example:

The sequence ${1/n}$ is bounded below by 0, since all terms are positive and approach 0 but never go below it.

Convergent Sequence

Criticality: 3

A sequence for which the limit of its terms as n approaches infinity exists and is a finite number.

Example:

The sequence ${1/n}$ is a convergent sequence because $\lim_{n \to \infty} (1/n) = 0$ , a finite value.

Convergent Series

Criticality: 3

A series for which the limit of its partial sums as n approaches infinity exists and is a finite number.

Example:

The series $\sum_{n=1}^{\infty} (1/n - 1/(n+1))$ is a convergent series because its sum is 1.

Decreasing Sequence

Criticality: 1

A sequence where each term is less than or equal to the previous term ($a_{n+1} \le a_n$ for all n).

Example:

The sequence ${1/n}$ (1, 1/2, 1/3, ...) is a decreasing sequence because each subsequent term is smaller.

Divergent Sequence

Criticality: 3

A sequence for which the limit of its terms as n approaches infinity does not exist or is infinite.

Example:

The sequence ${n^2}$ is a divergent sequence because $\lim_{n \to \infty} n^2 = \infty$ , which is not a finite value.

Divergent Series

Criticality: 3

A series for which the limit of its partial sums as n approaches infinity does not exist or is infinite.

Example:

The harmonic series $\sum_{n=1}^{\infty} 1/n$ is a divergent series even though its terms approach zero.

Harmonic Sequence

Criticality: 2

A specific type of sequence where each term is the reciprocal of a positive integer, typically represented as $\{1/n\}$.

Example:

The terms 1, 1/2, 1/3, 1/4, ... form a harmonic sequence, which is fundamental to understanding certain series behaviors.

Increasing Sequence

Criticality: 1

A sequence where each term is greater than or equal to the previous term ($a_{n+1} \ge a_n$ for all n).

Example:

The sequence ${n}$ (1, 2, 3, ...) is an increasing sequence as each term is larger than the one before it.

Infinite Series

Criticality: 3

A series where the sum extends indefinitely, represented as $s_\infty = \lim_{n \to \infty} \sum_{i=1}^{n} a_i$.

Example:

The sum of all positive integers, $1+2+3+...$ , is an infinite series.

Monotonic Sequence

Criticality: 2

A sequence that is either entirely increasing or entirely decreasing.

Example:

Both ${n}$ and ${1/n}$ are examples of a monotonic sequence because they consistently move in one direction.

Sequence

Criticality: 2

A sequence is an ordered list of terms, often related by a common pattern, represented as $\{a_n\}^\infty_1$.

Example:

The sequence ${1/n}^\infty_1$ generates terms like 1, 1/2, 1/3, ..., which is a classic example of a sequence.

Series

Criticality: 3

A series is the sum of the terms of a sequence.

Example:

The expression $1 + 1/2 + 1/3 + ...$ represents a series formed by summing the terms of the harmonic sequence.

Telescoping Series

Criticality: 3

A series where most of the terms cancel out in the partial sums, leaving only the first and last terms.

Example:

The series $\sum_{n=1}^{\infty} (1/n - 1/(n+1))$ is a telescoping series because its partial sums simplify to $1 - 1/(n+1)$ .

n-th Partial Sum

Criticality: 3

The sum of the first 'n' terms of a series, denoted as $s_n = \sum_{i=1}^{n} a_i$.

Example:

For the series $1 + 1/2 + 1/3 + ...$ , the 3rd partial sum ( $s_3$ ) would be $1 + 1/2 + 1/3 = 11/6$ .