Unit 10 Guide: Power, Taylor, and Maclaurin Series for AP Calculus BC

Taylor Polynomial

A finite polynomial used to approximate non-polynomial functions near a specific point.

$n$th-degree Taylor Polynomial

A polynomial of degree $n$ centered at $c$ used in Taylor approximation.

Linear Approximation

The first-degree Taylor polynomial matching the slope of the function at the center.

Maclaurin Series

A Taylor series centered at $c=0$.

Interval of Convergence (IOC)

The set of all $x$ values for which a power series converges.

Radius of Convergence

The distance within which a power series converges, denoted as $R$.

Lagrange Error Bound

A formula providing the worst-case error for Taylor polynomial approximations.

Power Series

An infinite series of the form $\sum a_n (x-c)^n$.

Derivative

The rate of change of a function with respect to its variable.

$P_n(x)$

The $n$th-degree Taylor polynomial of $f$ centered at $c$.

$f^{(n)}(c)$

The $n$th derivative of function $f$ evaluated at point $c$.

Convergence Tests

Methods like Alternating Series Test to determine the convergence of series.

Ratio Test

A test that determines convergence based on the limit of the ratio of consecutive terms.

Maximum value ($M$)

The maximum of the absolute value of the $(n+1)^{th}$ derivative on a given interval.

Odd Function

A function where $f(-x) = -f(x)$, like $ ext{sin}\,x$.

Even Function

A function where $f(-x) = f(x)$, like $ ext{cos}\,x$.

Geometric Series

A series of the form $\sum a r^n$, which converges for $|r|<1$.

Term-by-term differentiation

The process of differentiating a power series term by term within its interval of convergence.

First omitted term

In an alternating series, the first term not included in the approximation.

Convergence at Endpoints

The necessity to verify if a series converges or diverges at the endpoints of its interval.

Series Expansion

A representation of a function as an infinite sum of its derivatives at a single point.

Term structure in Taylor Series

The terms of a Taylor series are based on derivatives and factorial denominators.

Arithmetic error

Mistakes made in calculations, especially during series manipulation.

Function approximation

The process of estimating values of a function using simpler functions.

Symmetry in Functions

Properties of functions that show their behavior regarding the origin or y-axis.

Substitution method in series

A method to find new power series by substituting values into known series.