Taylor and Maclaurin Series Quiz

10 Questions

Who introduced the Taylor series in 1715?

Brook Taylor

What is the nth Taylor polynomial of a function?

A polynomial of degree n formed by the first n + 1 terms of the Taylor series

What do Taylor polynomials serve as for a function?

Approximations of the function

When is a Taylor series said to be convergent?

When its sum is the limit of the function

What does Taylor's theorem provide estimates on?

The error introduced by the use of Taylor polynomial approximations

Match the mathematician with the type of Taylor series named after them:

Brook Taylor = Taylor series Colin Maclaurin = Maclaurin series

Match the following with their role in Taylor series:

Taylor polynomial = Approximation of a function Taylor's theorem = Estimates the error in approximations Convergent Taylor series = Has a sum as the limit of the series Partial sum of Taylor series = Polynomial of degree n

Match the function with its property near the point of equality with the Taylor series:

Common functions = Equal to the sum of its Taylor series near a point Taylor series = Infinite sum of terms expressed in terms of derivatives at a single point

Match the following with their usage in Taylor series:

Taylor series = Infinite sum of terms expressed in terms of function's derivatives at a single point Taylor polynomials = Approximations of a function

Match the mathematician with their contribution to the Taylor series:

Brook Taylor = Introduced Taylor series in 1715 Colin Maclaurin = Made extensive use of Maclaurin series in the mid-18th century

Study Notes

History of Taylor Series

Brook Taylor introduced the Taylor series in 1715.

Taylor Polynomials

The nth Taylor polynomial of a function is an approximation of the function at a point.
Taylor polynomials serve as an approximation of a function around a point.

Convergence of Taylor Series

A Taylor series is said to be convergent if it converges to the original function at a point.

Taylor's Theorem

Taylor's theorem provides estimates on the accuracy of the Taylor polynomial approximations.

Mathematicians and Their Contributions

Brook Taylor: introduced the Taylor series in 1715.
Colin Maclaurin: named after the Maclaurin series, a special case of the Taylor series.

Key Concepts in Taylor Series

Center point: the point around which the Taylor series is expanded.
Lagrange remainder: used to estimate the error in approximating a function with its Taylor polynomial.

Properties of Functions and Taylor Series

Analytic function: a function that can be represented as a power series (i.e., a Taylor series) around a point.

Applications of Taylor Series

Approximation of functions: Taylor polynomials are used to approximate functions at a point.