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4. If a,z" following? has a radius of convergence R. what is the radius of convergence of each of the (a) an2 2.2m

4. If a,z" following? has a radius of convergence R. what is the radius of convergence of each of the (a) an2 2.2m

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Radius of Convergence for Power Series **4.** If \(\sum a_n z^n\) has a radius of convergence \(R\), what is the radius of convergence of each of the following? **(a)** \(\sum a_n z^{2n}\) **(b)** \(\sum a_n z^{n/2}\) **5.** Find the radius of convergence for each of the following series. --- **Explanation:** The problem involves determining the radius of convergence for variations of a given power series. Given that a power series \(\sum a_n z^n\) has a certain radius of convergence \(R\), you need to find out how the radius of convergence changes when the arguments of the series are altered. For part **(a)**, the argument is adjusted to \(z^{2n}\), and for part **(b)**, the argument is changed to \(z^{n/2}\). Understanding how transformations of the series' arguments affect the radius of convergence requires an understanding of series manipulations and transformations in complex analysis.
Transcribed Image Text:### Radius of Convergence for Power Series **4.** If \(\sum a_n z^n\) has a radius of convergence \(R\), what is the radius of convergence of each of the following? **(a)** \(\sum a_n z^{2n}\) **(b)** \(\sum a_n z^{n/2}\) **5.** Find the radius of convergence for each of the following series. --- **Explanation:** The problem involves determining the radius of convergence for variations of a given power series. Given that a power series \(\sum a_n z^n\) has a certain radius of convergence \(R\), you need to find out how the radius of convergence changes when the arguments of the series are altered. For part **(a)**, the argument is adjusted to \(z^{2n}\), and for part **(b)**, the argument is changed to \(z^{n/2}\). Understanding how transformations of the series' arguments affect the radius of convergence requires an understanding of series manipulations and transformations in complex analysis.
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