Chapter 8, Problem 9RCC

a.

To determine

To find: how do differentiate f and what is the radius of convergence of the series for f’ .

Answer to Problem 9RCC

The radius of convergence of the series for f’ is R .

Explanation of Solution

Given information: Suppose $f(x)$ is the sum of a power series with radius of

convergence R .

Calculation:

Suppose that $f(x)={\displaystyle \sum a_n}{(x-c)}^n.\text{Then}\text{have},$

  $f\text{'}(x)=\frac{df}{dx}=\frac{df}{dx}[{\displaystyle \sum a_n}{(x-c)}^n]={\displaystyle \sum a_n}\frac{df}{dx}{(x-c)}^n.$

The radius of convergence of the series for f’ is R .

b.

To determine

To find: how do integrate f and what the radius of convergence of the series is for ${\displaystyle \int f(x)dx}$ .

Answer to Problem 9RCC

The radius of convergence of the series for ${\displaystyle \int f(x)dx}$ is R .

Explanation of Solution

Given information: Suppose $f(x)$ is the sum of a power series with radius of

convergence R .

Calculation:

Suppose that $f(x)={\displaystyle \sum a_n}{(x-c)}^n.\text{Then}\text{have},$

  ${\displaystyle \int f(x)}dx={\displaystyle \int [{\displaystyle \sum a_n}{(x-c)}^n]}dx={\displaystyle \sum a_n}{\displaystyle \int {(x-c)}^{n}dx}.$

The radius of convergence of the series for ${\displaystyle \int f(x)dx}$ is R .