Glossary

Absolute Value (in context of Ratio Test)

Criticality: 2

The non-negative value of a real number, used in the Ratio Test to ensure the limit L is always non-negative, regardless of alternating signs in the series terms.

Example:

When calculating L for the Ratio Test, we take the absolute value of the ratio $\frac{a_{n+1}}{a_n}$ to handle potential negative terms and ensure the limit is positive.

Alternating Series Test

Criticality: 3

A test for series with alternating signs, stating that if the absolute value of the terms are decreasing and approach zero, the series converges.

Example:

The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ converges by the Alternating Series Test because $1/n$ is decreasing and approaches zero.

Convergence (of a series)

Criticality: 3

A series converges if its sequence of partial sums approaches a finite limit, meaning the sum of its infinite terms is a specific finite value.

Example:

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

Direct Comparison Test

Criticality: 2

A test that compares a given series to another series whose convergence or divergence is already known, based on term-by-term inequality.

Example:

To show $\sum \frac{1}{n^2+5}$ converges, you can use the Direct Comparison Test by comparing it to the convergent p-series $\sum \frac{1}{n^2}$ .

Divergence (of a series)

Criticality: 3

A series diverges if its sequence of partial sums does not approach a finite limit, often growing infinitely large, infinitely small, or oscillating without bound.

Example:

The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges, even though its terms approach zero.

Exponentials (in context of series)

Criticality: 2

Terms in a series where the variable appears in the exponent (e.g., $a^n$ or $n^a$), often making the Ratio Test a suitable choice for analysis due to cancellation properties.

Example:

The series $\sum \frac{3^n}{n!}$ contains an exponential term, $3^n$ , which signals the utility of the Ratio Test.

Factorials (in context of series)

Criticality: 2

The product of an integer and all positive integers below it, denoted by '!' (e.g., $n! = n \times (n-1) \times ... \times 1$), which simplifies nicely in the ratio of consecutive terms.

Example:

The term $n!$ in $\sum \frac{x^n}{n!}$ is a factorial, making the Ratio Test very effective due to cancellation properties like $(n+1)! = (n+1)n!$ .

Harmonic Series

Criticality: 2

A specific p-series where $p=1$, given by $\sum_{n=1}^\infty \frac{1}{n}$, which is a classic example of a divergent series.

Example:

The Harmonic Series is a classic example of a series whose terms approach zero but still diverges.

Indeterminate (for Ratio Test)

Criticality: 2

When the limit L in the Ratio Test equals 1, the test is *indeterminate*, meaning it provides no conclusion about the series' convergence or divergence, requiring another test.

Example:

If applying the Ratio Test to $\sum \frac{1}{n^2}$ yields L=1, the test is indeterminate, and you'd need the p-series test to confirm convergence.

Integral Test

Criticality: 2

A test that relates the convergence or divergence of a series to the convergence or divergence of an improper integral of a related continuous, positive, and decreasing function.

Example:

To determine if $\sum_{n=1}^\infty \frac{1}{n^2+1}$ converges, you could use the Integral Test by evaluating $\int_1^\infty \frac{1}{x^2+1} dx$ .

L'Hopital's Rule

Criticality: 2

A rule used to evaluate limits of indeterminate forms (like 0/0 or ∞/∞) by taking the derivatives of the numerator and denominator separately.

Example:

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

Limit Comparison Test

Criticality: 2

A test that compares two series by taking the limit of the ratio of their terms; if the limit is a finite, positive number, both series behave the same way.

Example:

For $\sum \frac{n}{n^2+1}$ , the Limit Comparison Test with $\sum \frac{1}{n}$ (harmonic series) shows it diverges.

Ratio Test

Criticality: 3

A test for series convergence that examines the limit L of the absolute ratio of consecutive terms, |a_(n+1)/a_n|. If L < 1, the series converges; if L > 1, it diverges; if L = 1, the test is indeterminate.

Example:

To determine if $\sum \frac{n^2}{2^n}$ converges, you'd apply the Ratio Test by evaluating $\lim_{n \to \infty} \left| \frac{(n+1)^2/2^{n+1}}{n^2/2^n} \right|$ .

Series (Σa_n)

Criticality: 3

The sum of the terms of an infinite sequence, represented as $\sum a_n$, where $a_n$ is the nth term of the sequence.

Example:

The expression $\sum_{n=1}^\infty \frac{1}{n^2}$ represents a series where each term is $a_n = \frac{1}{n^2}$ .

n-th term test

Criticality: 2

A test for divergence stating that if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges. If the limit is 0, the test is inconclusive.

Example:

For the series $\sum \frac{n}{n+1}$ , since $\lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0$ , the n-th term test tells us it diverges.

p-series test

Criticality: 3

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

Example:

The series $\sum_{n=1}^\infty \frac{1}{n^{1.5}}$ converges by the p-series test because $p=1.5 > 1$ .

Glossary

Absolute Value (in context of Ratio Test)

Criticality: 2

The non-negative value of a real number, used in the Ratio Test to ensure the limit L is always non-negative, regardless of alternating signs in the series terms.

Example:

When calculating L for the Ratio Test, we take the absolute value of the ratio $\frac{a_{n+1}}{a_n}$ to handle potential negative terms and ensure the limit is positive.

Alternating Series Test

Criticality: 3

A test for series with alternating signs, stating that if the absolute value of the terms are decreasing and approach zero, the series converges.

Example:

The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ converges by the Alternating Series Test because $1/n$ is decreasing and approaches zero.

Convergence (of a series)

Criticality: 3

A series converges if its sequence of partial sums approaches a finite limit, meaning the sum of its infinite terms is a specific finite value.

Example:

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

Direct Comparison Test

Criticality: 2

A test that compares a given series to another series whose convergence or divergence is already known, based on term-by-term inequality.

Example:

To show $\sum \frac{1}{n^2+5}$ converges, you can use the Direct Comparison Test by comparing it to the convergent p-series $\sum \frac{1}{n^2}$ .

Divergence (of a series)

Criticality: 3

A series diverges if its sequence of partial sums does not approach a finite limit, often growing infinitely large, infinitely small, or oscillating without bound.

Example:

The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges, even though its terms approach zero.

Exponentials (in context of series)

Criticality: 2

Terms in a series where the variable appears in the exponent (e.g., $a^n$ or $n^a$), often making the Ratio Test a suitable choice for analysis due to cancellation properties.

Example:

The series $\sum \frac{3^n}{n!}$ contains an exponential term, $3^n$ , which signals the utility of the Ratio Test.

Factorials (in context of series)

Criticality: 2

The product of an integer and all positive integers below it, denoted by '!' (e.g., $n! = n \times (n-1) \times ... \times 1$), which simplifies nicely in the ratio of consecutive terms.

Example:

The term $n!$ in $\sum \frac{x^n}{n!}$ is a factorial, making the Ratio Test very effective due to cancellation properties like $(n+1)! = (n+1)n!$ .

Harmonic Series

Criticality: 2

A specific p-series where $p=1$, given by $\sum_{n=1}^\infty \frac{1}{n}$, which is a classic example of a divergent series.

Example:

The Harmonic Series is a classic example of a series whose terms approach zero but still diverges.

Indeterminate (for Ratio Test)

Criticality: 2

When the limit L in the Ratio Test equals 1, the test is *indeterminate*, meaning it provides no conclusion about the series' convergence or divergence, requiring another test.

Example:

If applying the Ratio Test to $\sum \frac{1}{n^2}$ yields L=1, the test is indeterminate, and you'd need the p-series test to confirm convergence.

Integral Test

Criticality: 2

A test that relates the convergence or divergence of a series to the convergence or divergence of an improper integral of a related continuous, positive, and decreasing function.

Example:

To determine if $\sum_{n=1}^\infty \frac{1}{n^2+1}$ converges, you could use the Integral Test by evaluating $\int_1^\infty \frac{1}{x^2+1} dx$ .

L'Hopital's Rule

Criticality: 2

A rule used to evaluate limits of indeterminate forms (like 0/0 or ∞/∞) by taking the derivatives of the numerator and denominator separately.

Example:

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

Limit Comparison Test

Criticality: 2

A test that compares two series by taking the limit of the ratio of their terms; if the limit is a finite, positive number, both series behave the same way.

Example:

For $\sum \frac{n}{n^2+1}$ , the Limit Comparison Test with $\sum \frac{1}{n}$ (harmonic series) shows it diverges.

Ratio Test

Criticality: 3

Example:

To determine if $\sum \frac{n^2}{2^n}$ converges, you'd apply the Ratio Test by evaluating $\lim_{n \to \infty} \left| \frac{(n+1)^2/2^{n+1}}{n^2/2^n} \right|$ .

Series (Σa_n)

Criticality: 3

The sum of the terms of an infinite sequence, represented as $\sum a_n$, where $a_n$ is the nth term of the sequence.

Example:

The expression $\sum_{n=1}^\infty \frac{1}{n^2}$ represents a series where each term is $a_n = \frac{1}{n^2}$ .

n-th term test

Criticality: 2

A test for divergence stating that if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges. If the limit is 0, the test is inconclusive.

Example:

For the series $\sum \frac{n}{n+1}$ , since $\lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0$ , the n-th term test tells us it diverges.

p-series test

Criticality: 3

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

Example:

The series $\sum_{n=1}^\infty \frac{1}{n^{1.5}}$ converges by the p-series test because $p=1.5 > 1$ .