Power SeriesExpected Educational ResultsBloom’s TaxonomyPower SeriesRadius,

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

**Section 11.8**– Power Series**Section 11.9**– Representations of Functions as Power Series

**Objective 24–01**: I can use the*Geometric Series*to rewrite functions as power series.**Objective 24–02**: I can differentiate a power series written in summation notation.**Objective 24–03**: I can explain why the index changes when differentiating a power series written in summation notation.**Objective 24–04**: I can integrate a power series written in summation notation.**Objective 24–05**: I can use the geometric series to find the radius of convergence,*R*, of a power series.**Objective 24–06**: I can use the geometric series to find the interval of convergence,*I*, of a power series.

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.

**Recall**: A geometric series converges when:

**NOTE**: When the geometric series is used to find a power series, the radius,

**NOTE**: When the geometric series is used to find a power series, the interval of convergence, **always** an open interval:

**Example 01**: Find the radius,

**Solution**:

The power series is a geometric series since

According to the definition of a geometric series, the common ratio of the power series is

By the *Geometric Series Test*, the power series converges when

In interval notation, the interval of convergence is

The radius of convergence is

**NOTE**: On all Assessments, you **may** use technology to solve Absolute Value Inequalities without showing any work.

* Mathematica*:

`1``(* Example 01: Find the Interval of Convergence *)`

2`Reduce[RealAbs[3x^2]<1,x]`

**Warnings**:

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.You may need parens,

and$({\textstyle \phantom{\rule{0.167em}{0ex}}}$ , to group multiple terms in the numerator and denominator.$\phantom{\rule{0.167em}{0ex}}})$ *Reduce[ ]*has at least two arguments:The absolute value inequality to be solved;

The independent variable,

.$x$

For help on using the

*Reduce[ ]*function:In

*Mathematica*, execute the code:$\text{?Reduce}$ Click on

near the bottom-left of output$\vee $ Click on local

Read how to use the

*Reduce[ ]*function – you will be able to copy-paste code.

In Mathematica,

*Abs[3x^2]*is ;$\left|{\textstyle \phantom{\rule{0.167em}{0ex}}}3{x}^{2}{\textstyle \phantom{\rule{0.167em}{0ex}}}\right|$ *RealAbs[3x^2]*considers only real values of the argument.Remember, correct

*Mathematica*code will be all black except for variables.To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

** Mathematica solution**:

Use the geometric series to find the radius,

$f(x)=\frac{2}{1+x}$ $f(x)=\frac{3}{1-x}$ $f(x)=\frac{1}{x-1}$ $f(x)=\frac{1}{1-3{x}^{2}}$ $f(x)=\frac{1}{1+2{x}^{3}}$ $f(x)=\frac{x}{1-x}$ $f(x)=\frac{{x}^{3}}{1-{x}^{2}}$

When a power series converges at

, then$x=a$ .$R=0$ When a power series converges for all

, then$x$ .$R=\mathrm{\infty}$ When a power series converges for

then the radius of convergence is$R\in {\mathbb{R}}^{+}$ .$R$

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

**Section 11.8**: # 3, 5, 7, 9, 11, 13, 15, 19, 23, 25.

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 6:40 EDT
**Last Modified**: Saturday, 02 July 2022 - 09:13 (EDT)