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Glossary

A

Alternating Harmonic Series

Criticality: 2

A specific type of alternating series given by $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ or $\sum_{n=1}^\infty \frac{(-1)^n}{n}$, which is a key example for conditional convergence.

Example:

When you sum 1−12+13−14+…1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots, you are evaluating the Alternating Harmonic Series, which famously converges to ln⁡(2)\ln(2).

Alternating Series

Criticality: 3

A series whose terms alternate in sign, typically expressed in the form $\sum (-1)^n a_n$ or $\sum (-1)^{n+1} a_n$, where $a_n$ is a positive sequence.

Example:

The series 1−12+13−14+…1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots is a classic example of an alternating series.

Alternating Series Test

Criticality: 3

A theorem used to determine if an alternating series converges. It requires two conditions: the limit of the non-alternating part ($a_n$) must be zero, and $a_n$ must be decreasing.

Example:

To prove that ∑n=1∞(−1)nn\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}} converges, we apply the Alternating Series Test by verifying that lim⁡n→∞1n=0\lim_{n\to\infty} \frac{1}{\sqrt{n}} = 0 and that 1n\frac{1}{\sqrt{n}} is a decreasing sequence.

C

Convergence (of a series)

Criticality: 3

A series exhibits *convergence* if the sequence of its partial sums approaches a finite, specific value as the number of terms approaches infinity.

Example:

The geometric series ∑n=0∞(12)n\sum_{n=0}^\infty (\frac{1}{2})^n converges to 2, meaning its infinite sum is a finite number.

D

Divergence (of a series)

Criticality: 3

A series exhibits *divergence* if the sequence of its partial sums does not approach a finite limit, meaning the sum grows infinitely large, infinitely small, or oscillates without settling.

Example:

The harmonic series ∑n=1∞1n\sum_{n=1}^\infty \frac{1}{n} diverges, even though its individual terms approach zero, because its partial sums grow without bound.

a

a_n (in Alternating Series Test)

Criticality: 3

In the context of an alternating series $\sum (-1)^n a_n$ or $\sum (-1)^{n+1} a_n$, $a_n$ represents the positive, non-alternating sequence of terms that must satisfy the conditions of the Alternating Series Test.

Example:

For the series ∑n=1∞(−1)nn2+1\sum_{n=1}^\infty \frac{(-1)^n}{n^2+1}, the a_n term is 1n2+1\frac{1}{n^2+1}, which must be checked to see if its limit is zero and if it is decreasing.