Taylor Series

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.10 – Taylor and Maclaurin Series
• Section 11.11 – Applications of Taylor Polynomials

Expected Educational Results

• Objective 25–1: I can find the Taylor Series for any $f(x-a)$$f(x-a)$.
• Objective 25–2: I can find the Maclaurin Series for any $f(x)$$f(x)$.
• Objective 25–3: I can use the ratio test along with other series convergence tests to find the radius of convergence, $R$$R$, of a Taylor Series.
• Objective 25–4: I can use the ratio test along with other series convergence tests to find the interval of convergence, $I$$I$, of a Taylor Series.
• Objective 25–5: I can evaluate the sum of a Maclaurin Series using known common Maclaurin series.
• Objective 25–6: I can apply a Maclaurin Series to evaluating functions, limits, and integrals.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Taylor Series

Definition: Taylor Series, a.k.a., Taylor Polynomial

If $f$$f$ has a power series representation at $x=a$$x=a$, then the Taylor Series for a function, $f(x)$$f(x)$ is

${\displaystyle f(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n}$$\displaystyle f(x)=\sum_{n\,=\,0}^{\infty}{\frac{f^{\,(n)}(a)}{n!}(x-a)^n}$

NOTE: Taylor Series are Power Series.

Activity 09

Use the embedded DESMOS graph below to sketch functions and $n^{\text{th}}$$n^{\text{th}}$-degree (up to $n=10$$n=10$ Taylor Polynomials.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Sunday, 24 April 2022 - 19:55 (EDT)