Power Series Solutions

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

• Section 8.1 – Introduction – The Taylor Approximation
• Section 8.2 – Power Series and Analytic Functions
• Section 8.3 – Power Series Solutions to Linear Differential Equations
• Section 8.4 – Equations with Analytic Coefficients

Prerequisites

Definition: Power Function

A power function is a function in the form $x^n$.

Definition: Power Series

A power series is a sum of power functions: $\displaystyle \sum_{n\,=\,0}^{\infty}{c_n(x-x_0)^n}$ for some $x_0$ in some interval.

Definition: Power Series Centered at $x=a$

$\displaystyle\sum_{n=0}^{\infty}{c_n(x-a)^n}$

where $c_n$ are the coefficients of the power series.

Definition: Radius of Convergence of Power Series:

If $\displaystyle \lim_{n\rightarrow\infty}{\left|x-x_0\right|}=R$ where $R\geq 0$ is a positive real number, then the radius of convergence of the power series is $L$.

Definition: Interval of Convergence of a Power Series

The Interval of Convergence of a Power Series is the interval of all $x$-values for which the series converges.

Theorem

For a given power series $\displaystyle\sum_{n=0}^{\infty}{c_n(x-a)^n}$, the possible Interval of Convergence of a Power Series are

• The series converges only when $x=a$. [$R=0$].

• The series converges for all $x$. [$R=\infty$].

• There is a positive number $R$ such that the series converges if $|\,x-a\,|<R$ and diverges if $|\,x-a\,|>R$. NOTE: If $|\,x-a\,|=R$, then you need to verify the convergence at each endpoint, $a-R$ and $a+R$, leading to the following possible intervals of convergence:

  • $\big(a-R,a+R\big)$
  • $\big(a-R,a+R\big]$
  • $\big[a-R,a+R\big)$
  • $\big[a-R,a+R\big]$ where $R$ is the Radius of Convergence of the power series.

Investigation 01

1. Identify the type of series $1+x+x^2+x^3+x^4+\cdots$
2. Determine for what values of $x$, the series $1+x+x^2+x^3+x^4+\cdots$ converges. Explain.
3. Find the total sum of the convergent series. Explain.

Investigation 02

1. Rewrite the function $\displaystyle f(x)=\frac{1}{1-x}$ as a power series. Explain.

Investigation 03

Rewrite the following functions as power series:

1. $\displaystyle f(x)=\frac{1}{1-x^2}$
2. $\displaystyle f(x)=\frac{1}{1-x^3}$
3. $\displaystyle f(x)=\frac{1}{1+x}$
4. $\displaystyle f(x)=\frac{x}{1-x}$
5. $\displaystyle f(x)=\frac{x^2}{1+x}$
6. $\displaystyle f(x)=\frac{3}{1-2x}$
7. $\displaystyle f(x)=\frac{1}{3+x}$
8. $\displaystyle f(x)=\frac{1}{4-x^2}$

Investigation 04

1. Write out the first five (5) terms of $\displaystyle \sum_{n\,=\,0}^{\infty}{x^n}$
2. Differentiate the first five (5) terms of $\displaystyle \sum_{n\,=\,0}^{\infty}{x^n}$
3. Use the pattern to write the derivative from Step 2 as a series
4. Integrate the first five (5) terms of $\displaystyle \sum_{n\,=\,0}^{\infty}{x^n}$
5. Use the pattern to write the derivative from Step 4 as a series
6. Make a general conclusion about the differentiation and integration of power series.

Investigation 05

Rewrite the following functions as power series:

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

Series Convergence

Ratio Test

• A series converges if $\displaystyle \lim_{n\rightarrow\infty}\left|{\frac{a_{n+1}}{a_n}}\right|<1$
• A series diverges if $\displaystyle \lim_{n\rightarrow\infty}\left|{\frac{a_{n+1}}{a_n}}\right|>1$
• The ratio test is inconclusive if $\displaystyle \lim_{n\rightarrow\infty}\left|{\frac{a_{n+1}}{a_n}}\right|=1$

Investigation 06

Determine the interval and radius of convergence for each of the following power series:

1. Evaluate $\displaystyle \sum_{n\,=\,1}^{\infty}{\frac{2^n}{n}(4x-8)^n}$
2. Evaluate $\displaystyle \sum_{n\,=\,1}^{\infty}{\frac{(x-2)^n}{n5^n}}$
3. Evaluate $\displaystyle \sum_{n\,=\,1}^{\infty}{n!(2x+1)^n}$
4. Evaluate $\displaystyle \sum_{n\,=\,1}^{\infty}{(-1)^n\frac{n}{4^n}(x+3)^n}$

Differentiation of Power Series

Definition: Derivative of a Power Series

Suppose $\displaystyle f(x)=\sum_{n\,=\,0}^{\infty}{a_n(x-x_0)^n}$ is differentiable on its interval of convergence, then $\displaystyle f^{\,\prime}(x)=\sum_{n\,=\,1}^{\infty}{na_n(x-x_0)^{n-1}}$

Integration of Power Series

Definition: Integration of a Power Series

Suppose $\displaystyle f(x)=\sum_{n\,=\,0}^{\infty}{a_n(x-x_0)^n}$ is integrable on its interval of convergence, then

$\displaystyle \int{f(x)\,dx}=\sum_{n\,=\,0}^{\infty}{\frac{a_n}{n+1}(x-x_0)^{n+1}+C}$

Investigation 06

Find the derivative and antiderivative for each of the following power series:

1. $\displaystyle \sum_{n\,=\,0}^{\infty}{x^n}$
2. $\displaystyle \sum_{n\,=\,0}^{\infty}{\frac{(-1)^n}{(2n)!}x^{2n}}$

Shifting the Summation Index of a Power Series

Definition: Index Shift of a Power Series

Suppose $\displaystyle \sum_{n\,=\,i}^{\infty}{a_n x^n}$, then $\displaystyle \sum_{n\,=\,i-j}^{\infty}{a_{n+j} x^{n+j}}$

Investigation 07

Shift the following summation indices so that the series starts at $n=1$

1. $\displaystyle \sum_{n\,=\,2}^{\infty}{a_n n x^n}$
2. $\displaystyle \sum_{n\,=\,0}^{\infty}{\frac{(-1)^n n^3}{(n+2)!}x^{n}}$

Investigation 08

Shift the following summation indices to combine the following series [i.e., $x$ must have same exponent]:

1. $\displaystyle \sum_{n\,=\,1}^{\infty}{2n c_n x^{n-1}} + \sum_{n\,=\,0}^{\infty}{6c_n x^{n+1}}$

Taylor Series

Definition: Taylor Series

Suppose $f(x)$ is differentiable $n$ times at $x_0$, then the Taylor series, a.k.a., Taylor approximation, a.k.a., Taylor polynomial is

$\displaystyle \sum_{n\,=\,0}^{\infty}{\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n}$.

Maclaurin Series

Definition: Maclaurin Series

A Maclaurin series is a Taylor series when $\displaystyle x_0=0$.

Common Maclaurin Series

• $\displaystyle e^x=\sum_{n\,=\,0}^{\infty}{\frac{x^n}{n!}}$
• $\displaystyle \sin{(x)}=\sum_{n\,=\,0}^{\infty}{\frac{(-1)^n}{(2n+1)!}x^{2n+1}}$
• $\displaystyle \cos{(x)}=\sum_{n\,=\,0}^{\infty}{\frac{(-1)^n}{(2n)!}x^{2n}}$
• $\displaystyle \ln{(x)}=\sum_{n\,=\,1}^{\infty}{\frac{(-1)^{n-1}}{n}(x-1)^n}$

Odd and Even Numbers

An even number has the form $2n$ for any integer $n$, i.e., $n\in\mathbb{Z}$.

An odd number has the form $2n+1$ for any $n\in\mathbb{Z}$.

Product Notation

Here are some examples:

• $\displaystyle\prod_{i=1}^{1}{i}=1$
• $\displaystyle\prod_{i=1}^{4}{i}=1(2)(3)(4)$
• $\displaystyle\prod_{i=1}^{n}{i}=n!$
• $\displaystyle\prod_{i=1}^{3}{2i}=2(4)(6)$
• $\displaystyle\prod_{i=1}^{4}{i^2}=1(4)(9)(16)$

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Last Modified: Wednesday, 10 November 2020 18:26 EDT