Power SeriesExpected Educational ResultsBloom’s TaxonomyPower SeriesAlgebraDefinition: Power FunctionPower SeriesDefinition: Power SeriesDefinition: Power Series Centered at

**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.

A **power function** is a function with a single term that is the product of a real-valued coefficient, and a variable raised to a fixed real number. In other words, a **power function** contains a variable base raised to a fixed **power**.

**Example**:

A **power series** is a sum of power functions.

When all variables are subtracted by the same real number, e.g.,

Recall:

Use the geometric series to rewrite the following functions as a power series:

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

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

Use the geometric series to rewrite the following functions as a power series:

$f(x)=\frac{1}{1+x}$ $f(x)=\frac{4}{1+x}$ $f(x)=\frac{1}{1+{x}^{2}}$

Use the geometric series to rewrite the following functions as a power series:

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

Use the geometric series to rewrite the following functions as a power series:

$f(x)=\frac{1}{2-x}$ $f(x)=\frac{x}{3-{x}^{2}}$ $f(x)=\frac{{x}^{2}}{{x}^{2}-5}$

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**Created**: Tuesday, 1 December 2020 6:38 EDT
**Last Modified**: Monday, 15 November 2021 - 00:52 (EST)