# Power Series

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

• Section 11.8 – Power Series

• Section 11.9 – Representations of Functions as Power Series

## Expected Educational Results

• 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.

### 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.

## 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\left(x\right)=3{x}^{4}$$f(x)=3x^4$

$f\left(x\right)$$f(x)$ is a power function because $3{x}^{4}$$3x^4$ is the product of the real value, 3 [the coefficient], and ${x}^{4}$$x^4$ [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$$x=a$

$\sum _{n=0}^{\mathrm{\infty }}{c}_{n}\left(x-a{\right)}^{n}$$\sum_{n=0}^{\infty}{c_n(x-a)^n}$

#### Definition: Center of a Power Series

When all variables are subtracted by the same real number, e.g., $\left(x-a{\right)}^{n}$$(x-a)^n$, $a$$a$ horizontally shifts the graph of the function ${x}^{n}$$x^n$

#### Definition: Geometric Series

Recall: $\frac{1}{1-x}=1+x+{x}^{2}+{x}^{3}+\cdots =\sum _{n=0}^{\mathrm{\infty }}{x}^{n}$$\frac{1}{1-x}=1+x+x^2+x^3+\cdots=\sum_{n=0}^{\infty}{x^n}$ for $|\phantom{\rule{0.167em}{0ex}}x\phantom{\rule{0.167em}{0ex}}|<1$$|\,x\,|<1$.

#### Investigation 01

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

1. $f\left(x\right)=\frac{1}{1-x}$$\displaystyle f(x)=\frac{1}{1-x}$

2. $f\left(x\right)=\frac{3}{1-x}$$\displaystyle f(x)=\frac{3}{1-x}$

3. $f\left(x\right)=\frac{-2}{1-x}$$\displaystyle f(x)=\frac{-2}{1-x}$

#### Investigation 02

1. $f\left(x\right)=\frac{1}{1-2x}$$\displaystyle f(x)=\frac{1}{1-2x}$

2. $f\left(x\right)=\frac{3}{1-{x}^{2}}$$\displaystyle f(x)=\frac{3}{1-x^2}$

3. $f\left(x\right)=\frac{1}{1-2{x}^{3}}$$\displaystyle f(x)=\frac{1}{1-2x^3}$

#### Investigation 03

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

1. $f\left(x\right)=\frac{1}{1+x}$$\displaystyle f(x)=\frac{1}{1+x}$

2. $f\left(x\right)=\frac{4}{1+x}$$\displaystyle f(x)=\frac{4}{1+x}$

3. $f\left(x\right)=\frac{1}{1+{x}^{2}}$$\displaystyle f(x)=\frac{1}{1+x^2}$

#### Investigation 04

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

1. $f\left(x\right)=\frac{x}{1-x}$$\displaystyle f(x)=\frac{x}{1-x}$

2. $f\left(x\right)=\frac{{x}^{3}}{1-{x}^{2}}$$\displaystyle f(x)=\frac{x^3}{1-x^2}$

3. $f\left(x\right)=\frac{{x}^{2}}{1+{x}^{2}}$$\displaystyle f(x)=\frac{x^2}{1+x^2}$

#### Investigation 05

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

1. $f\left(x\right)=\frac{1}{2-x}$$\displaystyle f(x)=\frac{1}{2-x}$

2. $f\left(x\right)=\frac{x}{3-{x}^{2}}$$\displaystyle f(x)=\frac{x}{3-x^2}$

3. $f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-5}$$\displaystyle f(x)=\frac{x^2}{x^2-5}$