zuai-logo
zuai-logo
  2. AP Calculus
FlashcardFlashcardStudy GuideStudy GuideQuestion BankQuestion BankGlossaryGlossary

Glossary

C

Convergence

Criticality: 3

The property of an infinite series where its sequence of partial sums approaches a finite limit. For power series, this defines the specific interval or radius of x-values for which the series accurately represents the function.

Example:

When working with a geometric series like ∑n=0∞xn\sum_{n=0}^{\infty} x^n∑n=0∞​xn, we know it only achieves convergence when the absolute value of x is less than 1, meaning it sums to a finite value only within that range.

D

Derivative of a Power Series

Criticality: 3

The process of finding the derivative of a function represented by a power series by differentiating each term of the series individually, which results in a new power series.

Example:

If you have the power series for sin⁡(x)\sin(x)sin(x), taking the derivative of a power series term by term will yield the power series for cos⁡(x)\cos(x)cos(x), demonstrating a powerful connection between functions and their series representations.

G

General Term

Criticality: 3

The formula, often involving 'n', that defines the nth term of an infinite series, allowing the series to be written concisely using summation notation.

Example:

For the power series of sin⁡(x)\sin(x)sin(x), the general term is (−1)nx2n+1(2n+1)!\frac{(-1)^n x^{2n+1}}{(2n+1)!}(2n+1)!(−1)nx2n+1​, which helps us generate any term in the series by plugging in different values for n.

P

Power Series

Criticality: 3

An infinite series representation of a function, expressed as a sum of polynomial terms involving powers of (x-r), where r is the center of the series.

Example:

The power series for exe^xex is 1+x+x22!+x33!+…1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots1+x+2!x2​+3!x3​+…, which allows us to approximate exe^xex using a polynomial for any given x value. This is a fundamental example of a Power Series.