Chapter 10, Problem 72RE

(a)

To determine

The interval of convergence of the series $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{n{x}^n}{2^n}}$

Answer to Problem 72RE

The interval of convergence is $\left(-2,2\right)$

Explanation of Solution

Given information:

The given series is,

  $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{n{x}^n}{2^n}}$

Formula used:

Ratio test is used.

Calculation:

Check for absolute convergence using ratio test.

  $\begin{array}{l}\underset{n\to \infty }{\lim }\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\underset{n\to \infty }{\lim }\frac{n\left|x^n\right|}{2^n}\frac{2^{n-1}}{\left(n-1\right)\left|x^{n-1}\right|}\\ =\underset{n\to \infty }{\lim }\frac{n}{\left(n-1\right)}\frac{\left|x\right|}{2}\\ \underset{n\to \infty }{\lim }\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\left|x\right|}{2}\end{array}$

Hence,

  $\frac{\left|x\right|}{2}<1$

Therefore,

  $\left|x\right|<2$

That gives

  $-2<x<2$

Therefore, the interval of convergence is $\left(-2,2\right)$

Conclusion:

The interval of convergence is $\left(-2,2\right)$

(b)

To determine

The first nine terms that are sufficient to approximate f(-1)

Answer to Problem 72RE

The first nine terms are sufficient to approximate is $\frac{10}{1024}<0.01$

Explanation of Solution

Given information:

The given series is

  $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{n{\left(-1\right)}^n}{2^n}}$

Formula used:

The Alternating Series Estimation Theorem is used.

Calculation:

Putting $x=-1$

By the Alternating Series Estimation Theorem, since the size of the tenth term is $\frac{10}{1024}<0.01$

Therefore first nine terms is

  $\frac{10}{1024}<0.01$

Therefore first nine terms sufficient to approximate

Conclusion:

The first nine terms are sufficient to approximate is $\frac{10}{1024}<0.01$