Glossary

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 $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ , you are evaluating the Alternating Harmonic Series, which famously converges to $\ln(2)$ .

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 $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ 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 $\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}$ converges, we apply the Alternating Series Test by verifying that $\lim_{n\to\infty} \frac{1}{\sqrt{n}} = 0$ and that $\frac{1}{\sqrt{n}}$ is a decreasing sequence.

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 $\sum_{n=0}^\infty (\frac{1}{2})^n$ converges to 2, meaning its infinite sum is a finite number.

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

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

Glossary

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 $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ , you are evaluating the Alternating Harmonic Series, which famously converges to $\ln(2)$ .

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 $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ 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:

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 $\sum_{n=0}^\infty (\frac{1}{2})^n$ converges to 2, meaning its infinite sum is a finite number.

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

a_n (in Alternating Series Test)

Criticality: 3

Example:

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