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Chapter 8: Q. 61 (page 671)

Let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in $x$ with a finite radius of convergence $p$ . Prove that if the series converges absolutely at either $\pm p$ , then the series converges absolutely at the other value as well.

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01

Step 1. Given information.

given,

$\sum_{k = 0}^{\infty} a_{k} x^{k}$

02

Step 2.

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Most popular questions from this chapter

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .

$(1 + x)^{2 / 3}$

Use an appropriate Maclaurin series to find the values of the series in Exercises 17–22.

$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{2 k + 1} {(\frac{1}{\sqrt{3}})}^{2 k + 1}$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_{0}$ . Note: These are the same functions and values as in Exercises 41–48.

$53 . \ln (x), 3$

What is $x_{0}$ if the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ converges conditionally at both $x = - 4$ and $x = 8$ .

Prove that if $(- p, p]$ is the interval of convergence for the series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ , then the series converges conditionally at $p$ .

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