Power Series

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

Expected Educational Results

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.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

Power Series

Algebra

Definition: Power Function

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: f(x)=3x4

f(x) is a power function because 3x4 is the product of the real value, 3 [the coefficient], and x4 [a variable raised to a single real value].

Power Series

Definition: Power Series

A power series is a sum of power functions.

Definition: Power Series Centered at x=a

n=0cn(xa)n

Definition: Center of a Power Series

When all variables are subtracted by the same real number, e.g., (xa)n, a horizontally shifts the graph of the function xn

Definition: Geometric Series

Recall: 11x=1+x+x2+x3+=n=0xn for |x|<1.

Investigation 01

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

  1. f(x)=11x

  2. f(x)=31x

  3. f(x)=21x

Investigation 02

  1. f(x)=112x

  2. f(x)=31x2

  3. f(x)=112x3

Investigation 03

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

  1. f(x)=11+x

  2. f(x)=41+x

  3. f(x)=11+x2

Investigation 04

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

  1. f(x)=x1x

  2. f(x)=x31x2

  3. f(x)=x21+x2

Investigation 05

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

  1. f(x)=12x

  2. f(x)=x3x2

  3. f(x)=x2x25

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