Glossary

Comparison Tests for Convergence

Criticality: 3

Methods used to determine if an infinite series converges or diverges by comparing it to another series whose convergence or divergence is already known.

Example:

When faced with a complex series like $\sum \frac{1}{n^2+1}$ , you might use Comparison Tests for Convergence by comparing it to the simpler, known convergent series $\sum \frac{1}{n^2}$ .

Direct Comparison Test

Criticality: 3

A test for series $\sum a_n$ and $\sum b_n$ (where $a_n, b_n \geq 0$) stating that if $a_n \leq b_n$, then $\sum a_n$ converges if $\sum b_n$ converges, and $\sum b_n$ diverges if $\sum a_n$ diverges.

Example:

To show $\sum_{n=1}^\infty \frac{1}{n^3+5}$ converges, you can use the Direct Comparison Test by noting that $\frac{1}{n^3+5} < \frac{1}{n^3}$ , and $\sum \frac{1}{n^3}$ is a convergent p-series.

Geometric Series Test

Criticality: 3

A test for series of the form $\sum_{n=0}^\infty ar^n$. The series converges if $|r| < 1$ and diverges if $|r| \geq 1$.

Example:

The series $\sum_{n=0}^\infty 5 \left(\frac{1}{2}\right)^n$ converges because it is a Geometric Series Test with a common ratio $r = 1/2$ , which has an absolute value less than 1.

Harmonic Series

Criticality: 2

The divergent series $\sum_{n=1}^\infty \frac{1}{n}$. It is a special case of a p-series where $p=1$.

Example:

The series $\sum_{n=1}^\infty \frac{1}{n}$ is known as the Harmonic Series and is a classic example of a divergent series, even though its terms approach zero.

L'Hopital's Rule

Criticality: 2

A rule used to evaluate limits of indeterminate forms (like $\frac{0}{0}$ or $\frac{\infty}{\infty}$) by taking the derivatives of the numerator and denominator.

Example:

When evaluating $\lim_{x\to\infty} \frac{\ln x}{x}$ , you can apply L'Hopital's Rule to get $\lim_{x\to\infty} \frac{1/x}{1} = 0$ .

Limit Comparison Test

Criticality: 3

A test for series $\sum a_n$ and $\sum b_n$ (where $a_n, b_n \geq 0$) stating that if $\lim_{n\to\infty}\frac{a_n}{b_n} = L$ where $L$ is a positive, finite number, then both series either converge or diverge together.

Example:

To determine the convergence of $\sum_{n=1}^\infty \frac{n^2+1}{n^3-n}$ , you could use the Limit Comparison Test with $b_n = \frac{1}{n}$ , since the limit of their ratio as $n \to \infty$ is 1.

p-series test

Criticality: 3

A test for series of the form $\sum_{n=1}^\infty \frac{1}{n^p}$. The series converges if $p > 1$ and diverges if $0 < p \leq 1$.

Example:

The series $\sum_{n=1}^\infty \frac{1}{\sqrt{n}}$ diverges because it is a p-series test with $p = 1/2 \leq 1$ .

Glossary

Comparison Tests for Convergence

Criticality: 3

Methods used to determine if an infinite series converges or diverges by comparing it to another series whose convergence or divergence is already known.

Example:

When faced with a complex series like $\sum \frac{1}{n^2+1}$ , you might use Comparison Tests for Convergence by comparing it to the simpler, known convergent series $\sum \frac{1}{n^2}$ .

Direct Comparison Test

Criticality: 3

Example:

To show $\sum_{n=1}^\infty \frac{1}{n^3+5}$ converges, you can use the Direct Comparison Test by noting that $\frac{1}{n^3+5} < \frac{1}{n^3}$ , and $\sum \frac{1}{n^3}$ is a convergent p-series.

Geometric Series Test

Criticality: 3

A test for series of the form $\sum_{n=0}^\infty ar^n$. The series converges if $|r| < 1$ and diverges if $|r| \geq 1$.

Example:

The series $\sum_{n=0}^\infty 5 \left(\frac{1}{2}\right)^n$ converges because it is a Geometric Series Test with a common ratio $r = 1/2$ , which has an absolute value less than 1.

Harmonic Series

Criticality: 2

The divergent series $\sum_{n=1}^\infty \frac{1}{n}$. It is a special case of a p-series where $p=1$.

Example:

The series $\sum_{n=1}^\infty \frac{1}{n}$ is known as the Harmonic Series and is a classic example of a divergent series, even though its terms approach zero.

L'Hopital's Rule

Criticality: 2

A rule used to evaluate limits of indeterminate forms (like $\frac{0}{0}$ or $\frac{\infty}{\infty}$) by taking the derivatives of the numerator and denominator.

Example:

When evaluating $\lim_{x\to\infty} \frac{\ln x}{x}$ , you can apply L'Hopital's Rule to get $\lim_{x\to\infty} \frac{1/x}{1} = 0$ .

Limit Comparison Test

Criticality: 3

Example:

To determine the convergence of $\sum_{n=1}^\infty \frac{n^2+1}{n^3-n}$ , you could use the Limit Comparison Test with $b_n = \frac{1}{n}$ , since the limit of their ratio as $n \to \infty$ is 1.

p-series test

Criticality: 3

A test for series of the form $\sum_{n=1}^\infty \frac{1}{n^p}$. The series converges if $p > 1$ and diverges if $0 < p \leq 1$.

Example:

The series $\sum_{n=1}^\infty \frac{1}{\sqrt{n}}$ diverges because it is a p-series test with $p = 1/2 \leq 1$ .