Chapter 10.4, Problem 8E

To determine

Tofind:The radius of convergence of given power series.

Answer to Problem 8E

The radius of convergence of the series is $R=1$ .

Explanation of Solution

Given information:

The given power series is ${\displaystyle \sum _{n=0}^{\infty }{\left(x+5\right)}^n}$

Method used:

Let ${\displaystyle \sum _{n=0}^{\infty }{a}_n}$ be a given series, and let $r=\underset{n\to \infty }{\lim }\sqrt[n]{\left|a_n\right|}$ , then if $r<1$ the series ${\displaystyle \sum _{n=0}^{\infty }{a}_n}$ converges, if $r>1$ the series diverges, and if $r=1$ the root test for series inconclusive.

Let ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ be a given series. Then there exists a number R such that if $\left|x\right|<R$ the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ converges, if $\left|x\right|>R$ series diverges, the number R is called the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ .

The radius of convergence R of series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{\left(x-a\right)}^n$ is a non-negative number or $\infty$ such that the interval of convergence of power series is $\left[a-R,a+R\right]$ or $\left(a-R,a+R\right)$ .

Calculations:

The given series is ${\displaystyle \sum _{n=0}^{\infty }{\left(x+5\right)}^n}$ . We use ratio test and determine $r=\underset{n\to \infty }{\lim }\sqrt[n]{\left|a_n\right|}$ .

Here, $\Rightarrow r=\underset{n\to \infty }{\lim }{\left(\left|{\left(x+5\right)}^n\right|\right)}^{\frac{1}{n}}=\left|x+5\right|$ .

Thus, the series is convergent $r<1$

  $\Rightarrow \left|x+5\right|<1$

  $\Rightarrow -5-1<x<-5+1$

  $\Rightarrow -5-1<x<-5+1\approx c-R<x<c+R$

  $\Rightarrow R=1$

Hence, the radius of convergence of the series is $R=1$ .

Conclusion:

The radius of convergence of the series is $R=1$ .