Chapter 9.5, Problem 42E

a.

To determine

To find: The interval of convergence of the series.

Answer to Problem 42E

The interval of convergence of the series is $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=0}^{\infty }\frac{x^{2n+1}}{n!}}$

Calculation:

By applying the Ratio Test and looking at the interval's endpoints, one can determine the interval where the series will converge.

Taking the absolute value of the terms in the series and looking at the following limit, only using the Ratio Test for series with non-negative terms.

  $\begin{array}{c}\underset{n\to \infty }{\lim }\frac{\left|\frac{x^{2\left(n+1\right)+1}}{\left(n+1\right)!}\right|}{\left|\frac{x^{2n+1}}{n!}\right|}=\underset{n\to \infty }{\lim }\left(\frac{{\left|x\right|}^{2n+3}}{\left(n+1\right)!}\cdot \frac{n!}{{\left|x\right|}^{2n+1}}\right)\\ =\underset{n\to \infty }{\lim }\frac{{\left|x\right|}^2}{n+1}\\ =\infty \end{array}$

The series absolutely converges for every $x\in R$ based on the previous step.

Therefore, the interval of convergence of the series is $\left(-\infty ,\infty \right)$ .

b.

To determine

To find: For what value of $x$ does the series converge absolutely.

Answer to Problem 42E

The value of $x$ for which the series converges absolutely is $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=0}^{\infty }\frac{x^{2n+1}}{n!}}$

Calculation:

From part (a) it is known that,

The series absolutely converges for every $x$ based on the previous step.

Then, the series converges absolutely on $\left(-\infty ,\infty \right)$ .

Therefore, the value of $x$ for which the series converges absolutely is $\left(-\infty ,\infty \right)$ .

c.

To determine

To find: For what value of $x$ does the series converge conditionally.

Answer to Problem 42E

The value of $x$ for which the series converges conditionally does not exist.

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=0}^{\infty }\frac{x^{2n+1}}{n!}}$

Calculation:

There is no value of $x$ such that the series converges conditionally based on part (a), since the ratio test shows that the series converges absolutely on the interval $\left(-\infty ,\infty \right)$ .

Therefore, there does not exist a value of $x$ for which the series converges conditionally.