Elementary Differential Equations and Boundary Value Problems 9th Edition

Published by Wiley
ISBN 10: 0-47038-334-8
ISBN 13: 978-0-47038-334-6

Chapter 5 - Series Solutions of Second Order Linear Equations - 5.1 Review of Power Series - Problems - Page 249: 4

Answer

$$\rho = \frac 12$$

Work Step by Step

1. Use the ratio test to test for convergence: $$\lim_{n \longrightarrow \infty} \Bigg| \frac{ 2^{n+1}x^{n+1} }{2^nx^n} \Bigg | = \lim_{n \longrightarrow \infty} \Bigg| 2x \Bigg| = 2x$$ - Therefore, the series converges absolutely when $2x \lt 1$, or $x \lt \frac{1}{2}$ $$\rho = \frac 12$$
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