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}$

Definition: Maclaurin Series

A Maclaurin Series series is a Taylor Series for $x=0$$x=0$, that is ${\displaystyle f(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(0)}{n!}{x}^n}$$\displaystyle f(x)=\sum_{n\,=\,0}^{\infty}{\frac{f^{\,(n)}(0)}{n!}x^n}$

NOTE: Taylor Series and Maclaurin Series are Power Series.

Activity 01

Find the Maclaurin Series for $f(x)=e^{2x}$$f(x)=e^{2x}$ by completing the following table and finding a pattern: NOTE: In order to more easily find a pattern do not expand any expression.

Find the pattern in the last column and write $e^{2x}$$e^{2x}$ as a series.

Activity 02

Find the Maclaurin Series for $f(x)=e^{5x}$$f(x)=e^{5x}$ by completing the following table and finding a pattern: NOTE: In order to more easily find a pattern do not expand any expression.

Find the pattern in the last column to write $e^{5x}$$e^{5x}$ as a series.

NOTE: When completing the table, use exponents, e.g., $2^4$$2^4$ is more helpful than $16$$16$; do not rewrite factorials, e.g., $3!$$3!$ is more helpful than $6$$6$.

Investigation 03

From your work in Activities 1 and 2, what do you think is the Maclaurin Series for $f(x)=e^{bx}$$f(x)=e^{bx}$, $b>0$$b>0$? Explain.

Investigation 04

Find the Maclaurin Series for $f(x)=e^{x-1}$$f(x)=e^{x-1}$ by completing a similar table and finding a pattern.

Investigation 05

Find the Maclaurin Series for $f(x)=e^{-x}$$f(x)=e^{-x}$ by completing a similar table and finding a pattern.

Activity 06

Find the Maclaurin Series for $f(x)=\cos (3x)$$f(x)=\cos{(3x)}$ by completing a similar table and finding a pattern.

Activity 07

Find the Maclaurin Series for $f(x)=\sin (2x)$$f(x)=\sin{(2x)}$ by completing a similar table and finding a pattern.

Check Your Work

Use Technology to Verify a Taylor Series

NOTE: On all Assessments, you may use technology to verify your series convergence/divergence. You must show all Calculus II related work.

Mathematica

​x
1
(* Example from Investigation 09-1 *)
2
Series[Sin[x], {x,0,10}]
3

4
(* To remove the order of the error of the Taylor polynomial *)
5
Normal[Series[Sin[x], {x,0,10}]]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Series[ ] has at least two arguments:

   1. $f(x)$$f(x)$;

   2. $\{x,0,10\}$$\{x,0,10\}$ contains the following references

      • $x$$x$ is the independent variable;
      • $0$$0$ refers to the center; change this as needed;
      • $10$$10$ is the order (i.e., the highest exponent shown for the Taylor polynomial); change this as needed.

3. For help on using the Series[ ] function:

   1. In Mathematica, execute the code: $\text{?Series}$$\text{?Series}$
   2. Click on $\vee$$\or$ near the bottom-left of output
   3. Click on local
   4. Read how to use the Series[ ] function – you will be able to copy-paste code.

4. Mathematica will include $O(x^{11})$$O(x^{11})$ in the above result. This refers to the order of the error – you do not need to know about this.

5. Remember, correct Mathematica code will be all black except for variables.

6. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Investigation 08

Find the interval and radius of convergence for any of the Taylor Series from Investigation 07.

Homework

At this time, you should be able to complete the following assignments:

• Section 11.10: # 3, 4, 6, 7, 19, 24, 35, 36, 38, 45.

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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: Tuesday, 1 December 2020 15:05 EDT Last Modified: Monday, 15 November 2021 - 02:54 (EDT)