# 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 24–1: I can use the Geometric Series to rewrite functions as power series.

• Objective 24–2: I can differentiate a power series written in summation notation.

• Objective 24–3: I can explain why the index changes when differentiating a power series written in summation notation.

• Objective 24–4: I can integrate a power series written in summation notation.

• Objective 24–5: I can use the geometric series to find the radius of convergence, R, of a power series.

• Objective 24–6: 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

### Derivatives and Antiderivatives of Power Series

#### Definition: Derivative of a Power Series

$\frac{d}{dx}\left[\sum _{n=0}^{\mathrm{\infty }}{c}_{n}\left(x-a{\right)}^{n}\right]=\sum _{n=1}^{\mathrm{\infty }}n{c}_{n}\left(x-a{\right)}^{n-1}$$\displaystyle\dfrac{d}{dx}\left[\sum_{n=0}^{\infty}{c_n(x-a)^n}\right]=\sum_{n=1}^{\infty}{nc_n(x-a)^{n-1}}$

NOTE: Differentiating a power series shifts the index of the series.

#### Definition: Integral of a Power Series

$\int \left[\sum _{n=0}^{\mathrm{\infty }}{c}_{n}\left(x-a{\right)}^{n}\right]\phantom{\rule{0.167em}{0ex}}dx=C+\sum _{n=0}^{\mathrm{\infty }}{c}_{n}\frac{\left(x-a{\right)}^{n+1}}{n+1}$$\displaystyle\int{\left[\sum_{n=0}^{\infty}{c_n(x-a)^n}\right]\,dx}=C+\sum_{n=0}^{\infty}{c_n\frac{(x-a)^{n+1}}{n+1}}$

NOTE: Remember: Antiderivatives require the $+C$$+C$.

#### Investigation 07

Find the derivatives of the following power series:

1. $\sum _{n=1}^{\mathrm{\infty }}\frac{{x}^{n}}{\sqrt{n}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{x^n}{\sqrt{n}}}$

2. $\sum _{n=1}^{\mathrm{\infty }}\frac{{x}^{n}}{{n}^{3}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{x^n}{n^3}}$

3. $\sum _{n=0}^{\mathrm{\infty }}\left(-1{\right)}^{n}{4}^{n}{x}^{n}$$\displaystyle \sum_{n=0}^{\infty}{(-1)^n4^nx^n}$

4. $\sum _{n=0}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n}\left(x-3{\right)}^{n}}{2n+1}$$\displaystyle \sum_{n=0}^{\infty}{\frac{(-1)^n(x-3)^n}{2n+1}}$

5. $\sum _{n=1}^{\mathrm{\infty }}\frac{\left(2x-1{\right)}^{n}}{{5}^{n}\sqrt{n}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{(2x-1)^n}{5^n\sqrt{n}}}$

#### Investigation 08

Find the antiderivatives of the following power series:

1. $\sum _{n=1}^{\mathrm{\infty }}\frac{{x}^{n}}{\sqrt{n}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{x^n}{\sqrt{n}}}$

2. $\sum _{n=1}^{\mathrm{\infty }}\frac{{x}^{n}}{{n}^{3}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{x^n}{n^3}}$

3. $\sum _{n=0}^{\mathrm{\infty }}\left(-1{\right)}^{n}{4}^{n}{x}^{n}$$\displaystyle \sum_{n=0}^{\infty}{(-1)^n4^nx^n}$

4. $\sum _{n=0}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n}\left(x-3{\right)}^{n}}{2n+1}$$\displaystyle \sum_{n=0}^{\infty}{\frac{(-1)^n(x-3)^n}{2n+1}}$

5. $\sum _{n=1}^{\mathrm{\infty }}\frac{\left(2x-1{\right)}^{n}}{{5}^{n}\sqrt{n}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{(2x-1)^n}{5^n\sqrt{n}}}$

#### Investigation 09

Find the power series for ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$:

1. $f\left(x\right)={\mathrm{tan}}^{-1}\left(x\right)$$\displaystyle f(x)=\tan^{-1}(x)$

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

#### Investigation 10

Find the power series for $f\left(x\right)$$f(x)$:

1. $f\left(x\right)={\mathrm{tan}}^{-1}\left(x\right)$$\displaystyle f(x)=\tan^{-1}(x)$

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

#### Investigation 11

Find the power series for $\int f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\int{f(x)\,dx}$:

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

#### Investigation 12

Find the power series for $f\left(x\right)$$f(x)$:

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

### Homework

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

• Section 11.9: # 13, 17, 25.