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Q. 61

Expert-verified

Found in: Page 671

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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.

Ans:

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

given,

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

Step 2.

Most popular questions for Math Textbooks

Let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in x with an interval of convergence $[- 2, 2)$ . What is the radius of convergence of the power series $\sum_{k = 0}^{\infty} a_{k} (x - 3)^{k}$ ? Justify your answer.

What is $x_{0}$ if the interval of convergence for the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k} is (2, 10] ?$

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The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a nonnegative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x:

$J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

Graph the first four terms in the sequence of partial sums of $J_{1} (x)$ .

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Short Answer

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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Step 1. Given information.

Step 2.

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Let ∑k=0∞ akxk be a power series in x with a finite radius of convergence p. Prove that if the series converges absolutely at either ±p, then the series converges absolutely at the other value as well.

Step by Step Solution

Step 1. Given information.

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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.