Chapter 8.5, Problem 35E

To determine

To explain: Whether a power series has interval of convergence $\left[0,\infty \right)$ .

Answer to Problem 35E

No, it is impossible to find a power series whose interval of convergence is $\left[0,\infty \right)$ .

Explanation of Solution

Given information: Possibility of a power series with interval of convergence $\left[0,\infty \right)$ is to be established.

For a given power series ${\displaystyle \sum _{n=0}^{\infty }{c}_n{(x-a)}^n}$ , there are only three possibilities:

1. The series converges only when $x=a$ .

In this case, the interval of convergence is just a single point ‘ $a$ ’.

2. The series converges for all $x$ .

In this case, the interval of convergence is $\left(-\infty ,\infty \right)$ .

3. There is a positive number $R$ such that the series converges if $\left|x-a\right|<R$ and diverges if $\left|x-a\right|>R$ .

In this case, there are four possibilities for the interval of convergence:

$(a-R,a+R),(a-R,a+R],[a-R,a+R),[a-R,a+R]$ .

The interval of convergence is symmetric about the center of convergence with the exception of its endpoints.

Now, for the interval $\left[0,\infty \right)$ , a central number between $0$ and $\infty$ can’t be found.

Thus, it is not possible to find a power series whose interval of convergence is $\left[0,\infty \right)$ .