Chapter 8: Q. 16 (page 669)

Is it possible for a power series to have $(0, \infty)$ as its interval converge? Explain your answer.

Short Answer

Expert verified

If there is a positive real integer $ρ$ , the series will therefore absolutely converge for every $x \in (- ρ, ρ)$

Step by step solution

Step 1. Given information.

Given, Is the interval convergence of a power series ever $(0, \infty)$

Explanation

It is impossible for a power series with an x value to have an interval of convergence of $(0, \infty)$

The reason for this is if the power series $\sum_{i = 0}^{\infty} a_{k} x^{k}$ is the power series in x.

The series will therefore absolutely converge for every $x \in (- ρ, ρ)$ if there is a positive real number $ρ$ .

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Is it possible for a power series to have $(0, \infty)$ as its interval converge? Explain your answer.

Short Answer

Step by step solution

Step 1. Given information.

Explanation

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Is it possible for a power series to have (0,∞)as its interval converge? Explain your answer.

Short Answer

Step by step solution

Step 1. Given information.

Explanation

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Is it possible for a power series to have $(0, \infty)$ as its interval converge? Explain your answer.