Glossary

Converge

Criticality: 3

A series *converges* if the sum of its terms approaches a finite, specific value as the number of terms approaches infinity.

Example:

The series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ converges to 2, as its partial sums get closer and closer to 2.

Diverge

Criticality: 3

A series *diverges* if its sum does not approach a finite, specific value as the number of terms approaches infinity; instead, it grows infinitely large or oscillates.

Example:

If you keep adding terms of $1+2+3+4+\dots$ , the sum will continue to grow without bound, meaning the series diverges.

Evaluate the limit

Criticality: 3

The process of finding the specific value that a function or sequence approaches as its input or index tends towards a given point or infinity.

Example:

To determine if a series diverges using the nth term test, you must first evaluate the limit of its general term as n approaches infinity.

Limit notation

Criticality: 2

A mathematical way to express the value that a function or sequence approaches as its input or index gets arbitrarily close to a certain value, often infinity.

Example:

To describe the behavior of a function as x gets very large, we use limit notation like $\lim_{x \to \infty} f(x)$ .

Series

Criticality: 2

The sum of the terms of a sequence. In AP Calculus BC, this often refers to an infinite sum of terms.

Example:

The expression $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$ represents a series, specifically the harmonic series.

nth Term Test for Divergence

Criticality: 3

A test for infinite series stating that if the limit of the nth term as n approaches infinity is not zero, then the series diverges. If the limit is zero, the test is inconclusive.

Example:

When analyzing the series $\sum_{n=1}^\infty \frac{2n^2+1}{n^2-3}$ , applying the nth Term Test for Divergence reveals the limit of the terms is 2, so the series diverges.

Glossary

Converge

Criticality: 3

A series *converges* if the sum of its terms approaches a finite, specific value as the number of terms approaches infinity.

Example:

The series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ converges to 2, as its partial sums get closer and closer to 2.

Diverge

Criticality: 3

A series *diverges* if its sum does not approach a finite, specific value as the number of terms approaches infinity; instead, it grows infinitely large or oscillates.

Example:

If you keep adding terms of $1+2+3+4+\dots$ , the sum will continue to grow without bound, meaning the series diverges.

Evaluate the limit

Criticality: 3

The process of finding the specific value that a function or sequence approaches as its input or index tends towards a given point or infinity.

Example:

To determine if a series diverges using the nth term test, you must first evaluate the limit of its general term as n approaches infinity.

Limit notation

Criticality: 2

A mathematical way to express the value that a function or sequence approaches as its input or index gets arbitrarily close to a certain value, often infinity.

Example:

To describe the behavior of a function as x gets very large, we use limit notation like $\lim_{x \to \infty} f(x)$ .

Series

Criticality: 2

The sum of the terms of a sequence. In AP Calculus BC, this often refers to an infinite sum of terms.

Example:

The expression $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$ represents a series, specifically the harmonic series.

nth Term Test for Divergence

Criticality: 3

A test for infinite series stating that if the limit of the nth term as n approaches infinity is not zero, then the series diverges. If the limit is zero, the test is inconclusive.

Example:

When analyzing the series $\sum_{n=1}^\infty \frac{2n^2+1}{n^2-3}$ , applying the nth Term Test for Divergence reveals the limit of the terms is 2, so the series diverges.