Glossary

Absolute Convergence

Criticality: 3

A series $\sum a_n$ is absolutely convergent if the series formed by taking the absolute value of each term, $\sum |a_n|$, converges.

Example:

The series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ is absolutely convergent because $\sum_{n=1}^{\infty} \left|\frac{(-1)^n}{n^2}\right| = \sum_{n=1}^{\infty} \frac{1}{n^2}$ converges (it's a p-series with $p=2 > 1$ ).

Alternating Series Test

Criticality: 3

A test used to determine the convergence of an alternating series, which requires the terms to be positive, decreasing in magnitude, and approach zero as n approaches infinity.

Example:

To show that $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\ln(n+1)}$ converges, you would apply the Alternating Series Test, checking that $\frac{1}{\ln(n+1)}$ is positive, decreasing, and approaches zero.

Conditional Convergence

Criticality: 3

A series $\sum a_n$ is conditionally convergent if the series itself converges, but the series of its absolute values, $\sum |a_n|$, diverges.

Example:

The alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ is conditionally convergent because it converges by the Alternating Series Test, but its absolute value, $\sum_{n=1}^{\infty} \frac{1}{n}$ (the harmonic series), diverges.

Direct Comparison Test

Criticality: 2

A test for convergence or divergence of a series by comparing its terms to those of another series whose convergence or divergence is already known.

Example:

To determine if $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$ converges, you could use the Direct Comparison Test by comparing it to the convergent p-series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ .

Divergent

Criticality: 2

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

Example:

The series $\sum_{n=1}^{\infty} n$ is divergent because its terms continuously increase, causing the sum to grow without bound.

Harmonic Series

Criticality: 2

The series $\sum_{n=1}^{\infty} \frac{1}{n}$, which is a specific type of p-series where $p=1$.

Example:

When evaluating the absolute convergence of the alternating harmonic series, you'll encounter the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ , which is a classic example of a divergent series.

P-series

Criticality: 2

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, which converges if $p > 1$ and diverges if $p \leq 1$.

Example:

The series $\sum_{n=1}^{\infty} \frac{1}{n^4}$ is a p-series with $p=4$ , which immediately tells us it converges.

Glossary

Absolute Convergence

Criticality: 3

A series $\sum a_n$ is absolutely convergent if the series formed by taking the absolute value of each term, $\sum |a_n|$, converges.

Example:

Alternating Series Test

Criticality: 3

A test used to determine the convergence of an alternating series, which requires the terms to be positive, decreasing in magnitude, and approach zero as n approaches infinity.

Example:

Conditional Convergence

Criticality: 3

A series $\sum a_n$ is conditionally convergent if the series itself converges, but the series of its absolute values, $\sum |a_n|$, diverges.

Example:

Direct Comparison Test

Criticality: 2

A test for convergence or divergence of a series by comparing its terms to those of another series whose convergence or divergence is already known.

Example:

To determine if $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$ converges, you could use the Direct Comparison Test by comparing it to the convergent p-series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ .

Divergent

Criticality: 2

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

Example:

The series $\sum_{n=1}^{\infty} n$ is divergent because its terms continuously increase, causing the sum to grow without bound.

Harmonic Series

Criticality: 2

The series $\sum_{n=1}^{\infty} \frac{1}{n}$, which is a specific type of p-series where $p=1$.

Example:

P-series

Criticality: 2

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, which converges if $p > 1$ and diverges if $p \leq 1$.

Example:

The series $\sum_{n=1}^{\infty} \frac{1}{n^4}$ is a p-series with $p=4$ , which immediately tells us it converges.